Xenharmonic Wiki

User:Lériendil/Square and triangle superparticulars by prime subgroup: Difference between revisions

← Older edit
VisualWikitext
 
(40 intermediate revisions by the same user not shown)
Line 1: Line 1:
== 2-prime subgroup (2.3) ==
Some shorthand notation used here:
* S''k'' stands for ''k''^2/[(''k''-1)(''k''+1)] by standard convention (the ''k''th square superparticular).
* T''k'' = S''k'' * S(''k''+1) stands for [''k''(''k''+1)/2]/[(''k''-1)(''k''+2)/2] (the ''k''th triangle superparticular).
* U''k'' = S''k''/S(''k''+1) stands for the ''k''th ultraparticular, which has the same subgroup as T''k'' except in the case where k is congruent to 4 (mod 9), in which case the subgroup of U''k'' lacks prime 3 from that of T''k''.
* L''p'' refers to the ''p''-limit, i.e. the subgroup of primes less than or equal to ''p''.
* L''p''(-''q'') refers to the ''p'' limit with the prime ''q'' omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.
 
This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.
 
== 2- and 3-prime subgroups (2.3 and 2.3.p) ==
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.
 
=== 2-prime subgroup (2.3) ===


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! rowspan="2" | Superparticular
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! rowspan="2" | Subgroup
! colspan="2" | Comma
! colspan="2" | Comma
Line 9: Line 21:
! Ratio
! Ratio
! Smonzo
! Smonzo
|-
|- style="background-color: #BFD7FF;"
| T2
| [[3-limit|2.3]]
| [[3/2]]
| {{monzo| -1 1 }}
|-
| S2
| S2
| [[3-limit|2.3]]
| [[3-limit|2.3]]
| [[4/3]]
| [[4/3]]
| {{monzo| 2 -1 }}
| {{monzo| 2 -1 }}
|-
|- style="background-color: #BFD7FF;"
| S3
| S3
| [[3-limit|2.3]]
| [[3-limit|2.3]]
Line 26: Line 33:
|}
|}


== 3-prime subgroups (2.3.p) ==
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #BFD7FF;"
| T2
| [[3-limit|2.3]]
| [[3/2]]
| {{monzo| -1 1 }}
| U2
| [[3-limit|2.3]]
| [[32/27]]
| {{monzo| 5 -3 }}
|}
 
=== 3-prime subgroups (2.3.p) ===


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
! rowspan="2" | Superparticular
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! rowspan="2" | Subgroup
! colspan="2" | Comma
! colspan="2" | Comma
Line 36: Line 67:
! Ratio
! Ratio
! Smonzo
! Smonzo
|-
|- style="background-color: #FFBFEF;"
| T3
| [[5-limit|2.3.5]]
| [[6/5]]
| {{monzo| 1 1 -1 }}
|-
| T4
| [[5-limit|2.3.5]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
|-
| S4
| S4
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{monzo| 4 -1 -1 }}
|-
|- style="background-color: #FFBFEF;"
| S5
| S5
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{monzo| -3 -1 2 }}
|-
|- style="background-color: #FFBFEF;"
| S9
| S9
| [[5-limit|2.3.5]]
| [[5-limit|L5]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
|-
|- style="background-color: #C7BFDF;"
| T7
| [[2.3.7 subgroup|2.3.7]]
| [[28/27]]
| {{monzo| 2 -3 1 }}
|-
| S7
| S7
| [[2.3.7 subgroup|2.3.7]]
| [[2.3.7 subgroup|2.3.7]]
| [[49/48]]
| [[49/48]]
| {{monzo| -4 -1 2 }}
| {{monzo| -4 -1 2 }}
|-
|- style="background-color: #C7BFDF;"
| S8
| S8
| [[2.3.7 subgroup|2.3.7]]
| [[2.3.7 subgroup|2.3.7]]
| [[64/63]]
| [[64/63]]
| {{monzo| 6 -2 -1 }}
| {{monzo| 6 -2 -1 }}
|-
|- style="background-color: #F7F7BF;"
| S17
| S17
| 2.3.17
| 2.3.17
Line 82: Line 98:
| {{monzo| -5 -2 2 }}
| {{monzo| -5 -2 2 }}
|}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #FFBFEF;"
| T3
| [[5-limit|L5]]
| [[6/5]]
| {{monzo| 1 1 -1 }}
| U3
| [[5-limit|L5]]
| [[135/128]]
| {{monzo| -7 3 1 }}
|- style="background-color: #FFBFEF;"
| T4
| [[5-limit|L5]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| U4
| '''2.5'''
| [[128/125]]
| {{monzo| 7 -3 }}
|- style="background-color: #C7BFDF;"
| T7
| [[2.3.7 subgroup|2.3.7]]
| [[28/27]]
| {{monzo| 2 -3 1 }}
| U7
| [[2.3.7 subgroup|2.3.7]]
| [[1029/1024]]
| {{monzo| -10 1 3 }}
|}
== 4-prime subgroups ==
Note that the lists of triangle-particulars are ''complete'' and the insertion of higher primes will add no new inclusions to them. The lists of square particulars other than the "Higher primes" table are likewise complete.
=== 5-add-one-limit (L5.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #C7BFDF;"
| S6 = T8
| [[7-limit|L7]]
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
|- style="background-color: #C7BFDF;"
| S15
| [[7-limit|L7]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
|- style="background-color: #C7BFDF;"
| S49
| [[7-limit|L7]]
| [[2401/2400]]
| {{monzo| -5 -1 -2 4 }}
|- style="background-color: #F7C7BF;"
| S10
| L5.11
| [[100/99]]
| {{monzo| 2 -2 2 -1 }}
|- style="background-color: #F7C7BF;"
| S11
| L5.11
| [[121/120]]
| {{monzo| -3 -1 -1 2 }}
|- style="background-color: #D7BFEF;"
| S25
| L5.13
| [[625/624]]
| {{monzo| -4 -1 4 -1 }}
|- style="background-color: #D7BFEF;"
| S26
| L5.13
| [[676/675]]
| {{monzo| 2 -3 -2 2 }}
|- style="background-color: #F7F7BF;"
| S16
| L5.17
| [[256/255]]
| {{monzo| 8 -1 -1 -1 }}
|- style="background-color: #DFD7BF;"
| S19
| L5.19
| [[361/360]]
| {{monzo| -3 -2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S24
| L5.23
| [[576/575]]
| {{monzo| 6 2 -2 -1 }}
|- style="background-color: #CFC7F7;"
| S31
| L5.31
| [[961/960]]
| {{monzo| -6 -1 -1 2 }}
|- style="background-color: #FFBFEF;"
| S81
| L5.41
| [[6561/6560]]
| {{monzo| -5 8 -1 -1 }}
|- style="background-color: #FFBFEF;"
| S80
| L5.79
| [[6400/6399]]
| {{monzo| 8 -4 2 -1 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #C7BFDF;"
| T5
| [[7-limit|L7]]
| [[15/14]]
| {{monzo| -1 1 1 -1 }}
| U5
| [[7-limit|L7]]
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
|- style="background-color: #C7BFDF;"
| T6
| [[7-limit|L7]]
| [[21/20]]
| {{monzo| -2 1 -1 1 }}
| U6
| [[7-limit|L7]]
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
|- style="background-color: #C7BFDF;"
| T8 = S6
| [[7-limit|L7]]
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| U8
| [[7-limit|L7]]
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
|- style="background-color: #F7C7BF;"
| T9
| L5.11
| [[45/44]]
| {{monzo| -2 2 1 -1 }}
| U9
| L5.11
| [[8019/8000]]
| {{monzo| -6 6 -3 1 }}
|- style="background-color: #F7C7BF;"
| T10
| L5.11
| [[55/54]]
| {{monzo| -1 -3 1 1 }}
| U10
| L5.11
| [[4000/3993]]
| {{monzo| 5 -1 3 -3 }}
|- style="background-color: #D7BFEF;"
| T25
| L5.13
| [[325/324]]
| {{monzo| -2 -4 2 1 }}
| U25
| L5.13
| [[140625/140608]]
| {{monzo| -6 2 6 -3 }}
|- style="background-color: #F7F7BF;"
| T16
| L5.17
| [[136/135]]
| {{monzo| 3 -3 -1 1 }}
| U16
| L5.17
| [[24576/24565]]
| {{monzo| 13 1 -1 -3 }}
|}
=== 2.3.7.p subgroups ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S13
| 2.3.7.13
| [[169/168]]
| {{monzo| -3 -1 -1 2 }}
|- style="background-color: #D7BFEF;"
| S27
| 2.3.7.13
| [[729/728]]
| {{monzo| -3 6 -1 -1 }}
|- style="background-color: #D7DFCF;"
| S28
| 2.3.7.29
| [[784/783]]
| {{monzo| 4 -3 2 -1 }}
|- style="background-color: #CFC7F7;"
| S63
| 2.3.7.31
| [[3969/3968]]
| {{monzo| -7 4 2 -1 }}
|- style="background-color: #C7BFDF;"
| S48
| 2.3.7.47
| [[2304/2303]]
| {{monzo| 8 2 -2 -1 }}
|- style="background-color: #C7BFDF;"
| S97
| 2.3.7.97
| [[9409/9408]]
| {{monzo| -6 -1 -2 2 }}
|- style="background-color: #C7BFDF;"
| S127
| 2.3.7.127
| [[16129/16128]]
| {{monzo| -8 -2 -1 2 }}
|}
=== 2.3.11.p subgroups ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S12
| 2.3.11.13
| [[144/143]]
| {{monzo| 4 2 -1 -1 }}
|- style="background-color: #F7F7BF;"
| S33
| 2.3.11.17
| [[1089/1088]]
| {{monzo| -6 2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S23
| 2.3.11.23
| [[529/528]]
| {{monzo| -4 -1 -1 2 }}
|- style="background-color: #CFC7F7;"
| S32
| 2.3.11.31
| [[1024/1023]]
| {{monzo| 10 -1 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S243
| 2.3.11.61
| [[59049/59048]]
| {{monzo| -3 10 -2 -1 }}
|- style="background-color: #F7C7BF;"
| S242
| 2.3.11.241
| [[58564/58563]]
| {{monzo| 2 -5 4 -1 }}
|}
=== Higher primes ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S53
| 2.3.13.53
| [[2809/2808]]
| {{monzo| -3 -3 -1 2 }}
|- style="background-color: #DFD7BF;"
| S18
| 2.3.17.19
| [[324/323]]
| {{monzo| 2 4 -1 -1 }}
|- style="background-color: #F7F7BF;"
| S577
| 2.3.17.577
| [[332929/332928]]
| {{monzo| -7 -2 -2 2 }}
|- style="background-color: #DFD7BF;"
| S37
| 2.3.19.37
| [[1369/1368]]
| {{monzo| -3 -2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S47
| 2.3.23.47
| [[2209/2208]]
| {{monzo| -5 -1 -1 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #DFD7BF;"
| T17
| 2.3.17.19
| [[153/152]]
| {{monzo| -3 2 1 -1 }}
| U17
| 2.3.17.19
| [[93347/93312]]
| {{monzo| -7 -6 3 1 }}
|}
== 5-prime subgroups ==
=== 7-add-one-limit (L7.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #F7C7BF;"
| S21
| [[11-limit|L11]]
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
|- style="background-color: #F7C7BF;"
| S55
| [[11-limit|L11]]
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
|- style="background-color: #F7C7BF;"
| S99
| [[11-limit|L11]]
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
|- style="background-color: #D7BFEF;"
| S14
| L7.13
| [[196/195]]
| {{monzo| 2 -1 -1 2 -1 }}
|- style="background-color: #D7BFEF;"
| S64
| L7.13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 -1 }}
|- style="background-color: #F7F7BF;"
| S35 = T49
| L7.17
| [[1225/1224]]
| {{monzo| -3 -2 2 2 -1 }}
|- style="background-color: #F7F7BF;"
| S50
| L7.17
| [[2500/2499]]
| {{monzo| 2 -1 4 -2 -1 }}
|- style="background-color: #DFD7BF;"
| S20
| L7.19
| [[400/399]]
| {{monzo| 4 -1 2 -1 -1 }}
|- style="background-color: #FFDFBF;"
| S161
| L7.23
| [[25921/25920]]
| {{monzo| -6 -4 -1 2 2 }}
|- style="background-color: #D7DFCF;"
| S29
| L7.29
| [[841/840]]
| {{monzo| -3 -1 -1 -1 2 }}
|- style="background-color: #CFC7F7;"
| S125
| L7.31
| [[15625/15624]]
| {{monzo| -3 -2 6 -1 -1 }}
|- style="background-color: #C7BFDF;"
| S36
| L7.37
| [[1296/1295]]
| {{monzo| 4 4 -1 -1 -1 }}
|- style="background-color: #C7BFDF;"
| S41
| L7.41
| [[1681/1680]]
| {{monzo| -4 -1 -1 -1 2 }}
|- style="background-color: #C7BFDF;"
| S244
| L7.61
| [[59536/59535]]
| {{monzo| 4 -5 -1 -2 2 }}
|- style="background-color: #C7BFDF;"
| S71
| L7.71
| [[5041/5040]]
| {{monzo| -4 -2 -1 -1 2 }}
|- style="background-color: #C7BFDF;"
| S225
| L7.113
| [[50625/50624]]
| {{monzo| -6 4 -1 4 -1 }}
|- style="background-color: #C7BFDF;"
| S126
| L7.127
| [[15876/15875]]
| {{monzo| 2 4 -3 2 -1 }}
|- style="background-color: #C7BFDF;"
| S224
| L7.223
| [[50176/50175]]
| {{monzo| 10 -2 -2 2 -1 }}
|- style="background-color: #C7BFDF;"
| S251
| L7.251
| [[59536/59535]]
| {{monzo| -3 -2 -3 -1 2 }}
|- style="background-color: #C7BFDF;"
| S449
| L7.449
| [[201601/201600]]
| {{monzo| -7 -2 -2 -1 2 }}
|- style="background-color: #C7BFDF;"
| S4375
| L7.547
| [[19140625/19140624]]
| {{monzo| -4 -7 8 2 -1 }}
|- style="background-color: #C7BFDF;"
| S2401
| L7.1201
| [[5764801/5764800]]
| {{monzo| -5 -1 -2 8 -1 }}
|- style="background-color: #C7BFDF;"
| S2400
| L7.2399
| [[5760000/5759999]]
| {{monzo| 10 2 4 -4 -1 }}
|- style="background-color: #C7BFDF;"
| S4801
| L7.4801
| [[23049601/23049600]]
| {{monzo| -7 -1 -2 -4 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| T13
| L7.13
| [[91/90]]
| {{monzo| -1 -2 -1 1 1 }}
| U13
| '''2.5.7.13'''
| [[10985/10976]]
| {{monzo| -5 1 -3 3 }}
|- style="background-color: #D7BFEF;"
| T14
| L7.13
| [[105/104]]
| {{monzo| -3 1 1 1 -1 }}
| U14
| L7.13
| [[43904/43875]]
| {{monzo| 7 -3 -3 3 -1 }}
|- style="background-color: #D7BFEF;"
| T26
| L7.13
| [[351/350]]
| {{monzo| -1 3 -2 -1 1 }}
| U26
| L7.13
| [[492128/492075]]
| {{monzo| 5 -9 -2 1 3 }}
|- style="background-color: #F7F7BF;"
| T15
| L7.17
| [[120/119]]
| {{monzo| 3 1 1 -1 -1 }}
| U15
| L7.17
| [[57375/57344]]
| {{monzo| -13 3 3 -1 1 }}
|- style="background-color: #F7F7BF;"
| T49 = S35
| L7.17
| [[1225/1224]]
| {{monzo| -3 -2 2 2 -1 }}
| U49
| '''2.5.7.17'''
| [[2000033/2000000]]
| {{monzo| -7 -6 6 1 }}
|- style="background-color: #DFD7BF;"
| T19
| L7.19
| [[190/189]]
| {{monzo| 1 -3 1 -1 1 }}
| U19
| L7.19
| [[48013/48000]]
| {{monzo| -7 -1 -3 1 3 }}
|- style="background-color: #D7DFCF;"
| T28
| L7.29
| [[406/405]]
| {{monzo| 1 -4 -1 1 1 }}
| U28
| L7.29
| [[219520/219501]]
| {{monzo| 7 -2 1 3 -3 }}
|- style="background-color: #C7BFDF;"
| T126
| L7.127
| [[8001/8000]]
| {{monzo| -6 2 -3 1 1 }}
| U126
| L7.127
| [[256048128/256047875]]
| {{monzo| 10 6 -3 3 -3 }}
|}
=== 11-add-one-limit ===
==== L5.11.p subgroups ====
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| S65
| L13(-7)
| [[4225/4224]]
| {{monzo| -7 -1 2 -1 2 }}
|- style="background-color: #FFDFBF;"
| S45
| L5.11.23
| [[2025/2024]]
| {{monzo| -3 4 2 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S44
| L5.11.43
| [[1936/1935]]
| {{monzo| 4 -2 -1 2 -1 }}
|- style="background-color: #F7C7BF;"
| S54
| L5.11.53
| [[2916/2915]]
| {{monzo| 2 6 -1 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S121
| L5.11.61
| [[14641/14640]]
| {{monzo| -4 -1 -1 4 -1 }}
|- style="background-color: #F7C7BF;"
| S89
| L5.11.89
| [[7921/7920]]
| {{monzo| -4 -2 -1 -1 2 }}
|- style="background-color: #F7C7BF;"
| S485
| L5.11.97
| [[235225/235224]]
| {{monzo| -3 -5 2 -2 2 }}
|- style="background-color: #F7C7BF;"
| S100
| L5.11.101
| [[10000/9999]]
| {{monzo| 4 -2 4 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S109
| L5.11.109
| [[11881/11880]]
| {{monzo| -3 -3 -1 -1 2 }}
|- style="background-color: #F7C7BF;"
| S199
| L5.11.199
| [[39601/39600]]
| {{monzo| -4 -2 -2 -1 2 }}
|- style="background-color: #F7C7BF;"
| S241
| L5.11.241
| [[58081/58080]]
| {{monzo| -5 -1 -1 -2 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| T11
| L13(-7)
| [[66/65]]
| {{monzo| 1 1 -1 1 -1 }}
| U11
| L13(-7)
| [[17303/17280]]
| {{monzo| 7 3 1 -3 -1 }}
|- style="background-color: #FFDFBF;"
| T23
| L5.11.23
| [[276/275]]
| {{monzo| 2 1 -2 -1 1 }}
| U23
| L5.11.23
| [[304175/304128]]
| {{monzo| -10 -3 2 -1 3 }}
|- style="background-color: #CFC7F7;"
| T31
| L5.11.31
| [[496/495]]
| {{monzo| 4 -2 -1 -1 1 }}
| U31
| '''2.5.11.31'''
| [[327701/327680]]
| {{monzo| -16 -1 1 3 }}
|- style="background-color: #F7C7BF;"
| T241
| L5.11.241
| [[29161/29160]]
| {{monzo| -4 -6 -1 2 1 }}
| U241
| L5.11.241
| [[1133799201/1133799040]]
| {{monzo| -7 4 -1 -6 3 }}
|}
==== No-fives (L11(-5).p) subgroups ====
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Square-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|- style="background-color: #FFDFBF;"
| S22
| L11(-5).23
| [[484/483]]
| {{monzo| 2 -1 -1 2 -1 }}
|- style="background-color: #F7C7BF;"
| S43
| L11(-5).43
| [[1849/1848]]
| {{monzo| -3 -1 -1 -1 2 }}
|- style="background-color: #F7C7BF;"
| S98
| L11(-5).97
| [[9604/9603]]
| {{monzo| 2 -2 4 -1 -1 }}
|- style="background-color: #F7C7BF;"
| S197
| L11(-5).197
| [[38809/38808]]
| {{monzo| -3 -2 -2 -1 2 }}
|}
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Triangle-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Ultraparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|- style="background-color: #D7BFEF;"
| T12
| L13(-5)
| [[78/77]]
| {{monzo| 1 1 -1 -1 1 }}
| U12
| L13(-5)
| [[24192/24167]]
| {{monzo| 7 3 1 -1 -3 }}
|- style="background-color: #FFDFBF;"
| T22
| L11(-5).23
| [[253/252]]
| {{monzo| -2 -2 -1 1 1 }}
| U22
| '''2.7.11.23'''
| [[85184/85169]]
| {{monzo| 6 -1 3 -3 }}
|- style="background-color: #F7C7BF;"
| T97
| L11(-5).97
| [[4753/4752]]
| {{monzo| -4 -3 2 -1 1 }}
| U97
| L11(-5).97
| [[30118209/30118144]]
| {{monzo| -8 1 -6 1 3 }}
|}
== See also ==
* [[User:Lériendil/Third-superparticulars_and_semiparticulars_by_prime_subgroup|Third-superparticulars and semiparticulars by prime subgroup]]
Retrieved from "https://en.xen.wiki/w/User:Lériendil/Square_and_triangle_superparticulars_by_prime_subgroup"