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# Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*864b
if this polytope has a name.
Group : SmallGroup(864,1130)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 36, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,12,2} of size 1728
Vertex Figure Of :
{2,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*432a, {12,6}*432c
3-fold quotients : {12,12}*288b
4-fold quotients : {6,6}*216a
6-fold quotients : {6,12}*144b, {12,6}*144c
8-fold quotients : {6,3}*108
9-fold quotients : {4,12}*96a
12-fold quotients : {6,6}*72b
18-fold quotients : {2,12}*48, {4,6}*48a
24-fold quotients : {6,3}*36
27-fold quotients : {4,4}*32
36-fold quotients : {2,6}*24
54-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {2,3}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,12}*1728a, {12,12}*1728b, {24,12}*1728b, {12,24}*1728c, {12,24}*1728e
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)
(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)
(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)
(133,157)(134,159)(135,158)(163,190)(164,192)(165,191)(166,196)(167,198)
(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)(175,205)
(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)(183,209)
(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);;
s1 := (  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)(  8,116)
(  9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)( 16,133)
( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)( 24,120)
( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)
( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)( 40,154)
( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)( 48,150)
( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,166)( 56,167)
( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)( 64,184)
( 65,185)( 66,186)( 67,181)( 68,182)( 69,183)( 70,187)( 71,188)( 72,189)
( 73,175)( 74,176)( 75,177)( 76,172)( 77,173)( 78,174)( 79,178)( 80,179)
( 81,180)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)( 88,196)
( 89,197)( 90,198)( 91,211)( 92,212)( 93,213)( 94,208)( 95,209)( 96,210)
( 97,214)( 98,215)( 99,216)(100,202)(101,203)(102,204)(103,199)(104,200)
(105,201)(106,205)(107,206)(108,207);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)(  8, 15)
(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)
( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)
( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107)
(109,199)(110,200)(111,201)(112,207)(113,205)(114,206)(115,203)(116,204)
(117,202)(118,190)(119,191)(120,192)(121,198)(122,196)(123,197)(124,194)
(125,195)(126,193)(127,208)(128,209)(129,210)(130,216)(131,214)(132,215)
(133,212)(134,213)(135,211)(136,172)(137,173)(138,174)(139,180)(140,178)
(141,179)(142,176)(143,177)(144,175)(145,163)(146,164)(147,165)(148,171)
(149,169)(150,170)(151,167)(152,168)(153,166)(154,181)(155,182)(156,183)
(157,189)(158,187)(159,188)(160,185)(161,186)(162,184);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(216)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)
(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)
(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)
(132,161)(133,157)(134,159)(135,158)(163,190)(164,192)(165,191)(166,196)
(167,198)(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)
(175,205)(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)
(183,209)(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);
s1 := Sym(216)!(  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)
(  8,116)(  9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)
( 16,133)( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)
( 24,120)( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)
( 32,137)( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)
( 40,154)( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)
( 48,150)( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,166)
( 56,167)( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)
( 64,184)( 65,185)( 66,186)( 67,181)( 68,182)( 69,183)( 70,187)( 71,188)
( 72,189)( 73,175)( 74,176)( 75,177)( 76,172)( 77,173)( 78,174)( 79,178)
( 80,179)( 81,180)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)
( 88,196)( 89,197)( 90,198)( 91,211)( 92,212)( 93,213)( 94,208)( 95,209)
( 96,210)( 97,214)( 98,215)( 99,216)(100,202)(101,203)(102,204)(103,199)
(104,200)(105,201)(106,205)(107,206)(108,207);
s2 := Sym(216)!(  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)
(  8, 15)(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)
( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)
( 51, 53)( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)
( 62, 69)( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)
( 85, 99)( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)
(105,107)(109,199)(110,200)(111,201)(112,207)(113,205)(114,206)(115,203)
(116,204)(117,202)(118,190)(119,191)(120,192)(121,198)(122,196)(123,197)
(124,194)(125,195)(126,193)(127,208)(128,209)(129,210)(130,216)(131,214)
(132,215)(133,212)(134,213)(135,211)(136,172)(137,173)(138,174)(139,180)
(140,178)(141,179)(142,176)(143,177)(144,175)(145,163)(146,164)(147,165)
(148,171)(149,169)(150,170)(151,167)(152,168)(153,166)(154,181)(155,182)
(156,183)(157,189)(158,187)(159,188)(160,185)(161,186)(162,184);
poly := sub<Sym(216)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope