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# Polytope of Type {12,12}

Atlas Canonical Name : {12,12}*1728s
if this polytope has a name.
Group : SmallGroup(1728,33548)
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
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,12}*864k
3-fold quotients : {4,12}*576
4-fold quotients : {12,6}*432i
6-fold quotients : {4,12}*288
8-fold quotients : {12,6}*216c
9-fold quotients : {12,4}*192a
12-fold quotients : {4,6}*144
18-fold quotients : {12,4}*96a
24-fold quotients : {4,6}*72
27-fold quotients : {4,4}*64
36-fold quotients : {12,2}*48, {6,4}*48a
54-fold quotients : {4,4}*32
72-fold quotients : {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  8)(  3,  6)(  4,  7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)( 14, 23)
( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)( 37, 46)
( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)( 45, 54)
( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)( 60, 84)( 61, 85)( 62, 83)
( 63, 90)( 64,100)( 65,107)( 66,105)( 67,106)( 68,104)( 69,102)( 70,103)
( 71,101)( 72,108)( 73, 91)( 74, 98)( 75, 96)( 76, 97)( 77, 95)( 78, 93)
( 79, 94)( 80, 92)( 81, 99)(110,116)(111,114)(112,115)(118,127)(119,134)
(120,132)(121,133)(122,131)(123,129)(124,130)(125,128)(126,135)(137,143)
(138,141)(139,142)(145,154)(146,161)(147,159)(148,160)(149,158)(150,156)
(151,157)(152,155)(153,162)(163,190)(164,197)(165,195)(166,196)(167,194)
(168,192)(169,193)(170,191)(171,198)(172,208)(173,215)(174,213)(175,214)
(176,212)(177,210)(178,211)(179,209)(180,216)(181,199)(182,206)(183,204)
(184,205)(185,203)(186,201)(187,202)(188,200)(189,207);;
s1 := (  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)
(109,172)(110,173)(111,174)(112,180)(113,178)(114,179)(115,176)(116,177)
(117,175)(118,163)(119,164)(120,165)(121,171)(122,169)(123,170)(124,167)
(125,168)(126,166)(127,181)(128,182)(129,183)(130,189)(131,187)(132,188)
(133,185)(134,186)(135,184)(136,199)(137,200)(138,201)(139,207)(140,205)
(141,206)(142,203)(143,204)(144,202)(145,190)(146,191)(147,192)(148,198)
(149,196)(150,197)(151,194)(152,195)(153,193)(154,208)(155,209)(156,210)
(157,216)(158,214)(159,215)(160,212)(161,213)(162,211);;
s2 := (  1,140)(  2,139)(  3,141)(  4,137)(  5,136)(  6,138)(  7,143)(  8,142)
(  9,144)( 10,149)( 11,148)( 12,150)( 13,146)( 14,145)( 15,147)( 16,152)
( 17,151)( 18,153)( 19,158)( 20,157)( 21,159)( 22,155)( 23,154)( 24,156)
( 25,161)( 26,160)( 27,162)( 28,113)( 29,112)( 30,114)( 31,110)( 32,109)
( 33,111)( 34,116)( 35,115)( 36,117)( 37,122)( 38,121)( 39,123)( 40,119)
( 41,118)( 42,120)( 43,125)( 44,124)( 45,126)( 46,131)( 47,130)( 48,132)
( 49,128)( 50,127)( 51,129)( 52,134)( 53,133)( 54,135)( 55,194)( 56,193)
( 57,195)( 58,191)( 59,190)( 60,192)( 61,197)( 62,196)( 63,198)( 64,203)
( 65,202)( 66,204)( 67,200)( 68,199)( 69,201)( 70,206)( 71,205)( 72,207)
( 73,212)( 74,211)( 75,213)( 76,209)( 77,208)( 78,210)( 79,215)( 80,214)
( 81,216)( 82,167)( 83,166)( 84,168)( 85,164)( 86,163)( 87,165)( 88,170)
( 89,169)( 90,171)( 91,176)( 92,175)( 93,177)( 94,173)( 95,172)( 96,174)
( 97,179)( 98,178)( 99,180)(100,185)(101,184)(102,186)(103,182)(104,181)
(105,183)(106,188)(107,187)(108,189);;
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*s0*s1*s2*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,  8)(  3,  6)(  4,  7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)
( 14, 23)( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)
( 37, 46)( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)
( 45, 54)( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)( 60, 84)( 61, 85)
( 62, 83)( 63, 90)( 64,100)( 65,107)( 66,105)( 67,106)( 68,104)( 69,102)
( 70,103)( 71,101)( 72,108)( 73, 91)( 74, 98)( 75, 96)( 76, 97)( 77, 95)
( 78, 93)( 79, 94)( 80, 92)( 81, 99)(110,116)(111,114)(112,115)(118,127)
(119,134)(120,132)(121,133)(122,131)(123,129)(124,130)(125,128)(126,135)
(137,143)(138,141)(139,142)(145,154)(146,161)(147,159)(148,160)(149,158)
(150,156)(151,157)(152,155)(153,162)(163,190)(164,197)(165,195)(166,196)
(167,194)(168,192)(169,193)(170,191)(171,198)(172,208)(173,215)(174,213)
(175,214)(176,212)(177,210)(178,211)(179,209)(180,216)(181,199)(182,206)
(183,204)(184,205)(185,203)(186,201)(187,202)(188,200)(189,207);
s1 := 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)(109,172)(110,173)(111,174)(112,180)(113,178)(114,179)(115,176)
(116,177)(117,175)(118,163)(119,164)(120,165)(121,171)(122,169)(123,170)
(124,167)(125,168)(126,166)(127,181)(128,182)(129,183)(130,189)(131,187)
(132,188)(133,185)(134,186)(135,184)(136,199)(137,200)(138,201)(139,207)
(140,205)(141,206)(142,203)(143,204)(144,202)(145,190)(146,191)(147,192)
(148,198)(149,196)(150,197)(151,194)(152,195)(153,193)(154,208)(155,209)
(156,210)(157,216)(158,214)(159,215)(160,212)(161,213)(162,211);
s2 := Sym(216)!(  1,140)(  2,139)(  3,141)(  4,137)(  5,136)(  6,138)(  7,143)
(  8,142)(  9,144)( 10,149)( 11,148)( 12,150)( 13,146)( 14,145)( 15,147)
( 16,152)( 17,151)( 18,153)( 19,158)( 20,157)( 21,159)( 22,155)( 23,154)
( 24,156)( 25,161)( 26,160)( 27,162)( 28,113)( 29,112)( 30,114)( 31,110)
( 32,109)( 33,111)( 34,116)( 35,115)( 36,117)( 37,122)( 38,121)( 39,123)
( 40,119)( 41,118)( 42,120)( 43,125)( 44,124)( 45,126)( 46,131)( 47,130)
( 48,132)( 49,128)( 50,127)( 51,129)( 52,134)( 53,133)( 54,135)( 55,194)
( 56,193)( 57,195)( 58,191)( 59,190)( 60,192)( 61,197)( 62,196)( 63,198)
( 64,203)( 65,202)( 66,204)( 67,200)( 68,199)( 69,201)( 70,206)( 71,205)
( 72,207)( 73,212)( 74,211)( 75,213)( 76,209)( 77,208)( 78,210)( 79,215)
( 80,214)( 81,216)( 82,167)( 83,166)( 84,168)( 85,164)( 86,163)( 87,165)
( 88,170)( 89,169)( 90,171)( 91,176)( 92,175)( 93,177)( 94,173)( 95,172)
( 96,174)( 97,179)( 98,178)( 99,180)(100,185)(101,184)(102,186)(103,182)
(104,181)(105,183)(106,188)(107,187)(108,189);
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*s0*s1*s2*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