# Polytope of Type {12,12}

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

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

```
References : None.
to this polytope