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# Polytope of Type {12,12}

Atlas Canonical Name : {12,12}*1728g
if this polytope has a name.
Group : SmallGroup(1728,12630)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,12}*864g
3-fold quotients : {12,4}*576
4-fold quotients : {6,12}*432e
6-fold quotients : {12,4}*288
8-fold quotients : {6,12}*216a
12-fold quotients : {6,4}*144
24-fold quotients : {6,4}*72
27-fold quotients : {4,4}*64
54-fold quotients : {4,4}*32
108-fold quotients : {2,4}*16, {4,2}*16
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,136)(  2,137)(  3,138)(  4,144)(  5,142)(  6,143)(  7,140)(  8,141)
(  9,139)( 10,156)( 11,154)( 12,155)( 13,161)( 14,162)( 15,160)( 16,157)
( 17,158)( 18,159)( 19,146)( 20,147)( 21,145)( 22,151)( 23,152)( 24,153)
( 25,150)( 26,148)( 27,149)( 28,109)( 29,110)( 30,111)( 31,117)( 32,115)
( 33,116)( 34,113)( 35,114)( 36,112)( 37,129)( 38,127)( 39,128)( 40,134)
( 41,135)( 42,133)( 43,130)( 44,131)( 45,132)( 46,119)( 47,120)( 48,118)
( 49,124)( 50,125)( 51,126)( 52,123)( 53,121)( 54,122)( 55,190)( 56,191)
( 57,192)( 58,198)( 59,196)( 60,197)( 61,194)( 62,195)( 63,193)( 64,210)
( 65,208)( 66,209)( 67,215)( 68,216)( 69,214)( 70,211)( 71,212)( 72,213)
( 73,200)( 74,201)( 75,199)( 76,205)( 77,206)( 78,207)( 79,204)( 80,202)
( 81,203)( 82,163)( 83,164)( 84,165)( 85,171)( 86,169)( 87,170)( 88,167)
( 89,168)( 90,166)( 91,183)( 92,181)( 93,182)( 94,188)( 95,189)( 96,187)
( 97,184)( 98,185)( 99,186)(100,173)(101,174)(102,172)(103,178)(104,179)
(105,180)(106,177)(107,175)(108,176);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)
( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 72)( 65, 71)( 66, 70)( 67, 69)
( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,102)(103,108)(104,107)(105,106)
(109,193)(110,195)(111,194)(112,190)(113,192)(114,191)(115,196)(116,198)
(117,197)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)
(125,200)(126,199)(127,210)(128,209)(129,208)(130,216)(131,215)(132,214)
(133,213)(134,212)(135,211)(136,166)(137,168)(138,167)(139,163)(140,165)
(141,164)(142,169)(143,171)(144,170)(145,180)(146,179)(147,178)(148,177)
(149,176)(150,175)(151,174)(152,173)(153,172)(154,183)(155,182)(156,181)
(157,189)(158,188)(159,187)(160,186)(161,185)(162,184);;
s2 := (  1,  2)(  4, 23)(  5, 22)(  6, 24)(  7, 16)(  8, 18)(  9, 17)( 10, 20)
( 11, 19)( 12, 21)( 13, 14)( 26, 27)( 28, 29)( 31, 50)( 32, 49)( 33, 51)
( 34, 43)( 35, 45)( 36, 44)( 37, 47)( 38, 46)( 39, 48)( 40, 41)( 53, 54)
( 55, 83)( 56, 82)( 57, 84)( 58,104)( 59,103)( 60,105)( 61, 97)( 62, 99)
( 63, 98)( 64,101)( 65,100)( 66,102)( 67, 95)( 68, 94)( 69, 96)( 70, 88)
( 71, 90)( 72, 89)( 73, 92)( 74, 91)( 75, 93)( 76, 86)( 77, 85)( 78, 87)
( 79,106)( 80,108)( 81,107)(109,110)(112,131)(113,130)(114,132)(115,124)
(116,126)(117,125)(118,128)(119,127)(120,129)(121,122)(134,135)(136,137)
(139,158)(140,157)(141,159)(142,151)(143,153)(144,152)(145,155)(146,154)
(147,156)(148,149)(161,162)(163,191)(164,190)(165,192)(166,212)(167,211)
(168,213)(169,205)(170,207)(171,206)(172,209)(173,208)(174,210)(175,203)
(176,202)(177,204)(178,196)(179,198)(180,197)(181,200)(182,199)(183,201)
(184,194)(185,193)(186,195)(187,214)(188,216)(189,215);;
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, s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*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*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope