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# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*864b
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 72, 216, 36
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,6,2} of size 1728
Vertex Figure Of :
{2,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6}*432d
3-fold quotients : {12,6}*288a
4-fold quotients : {6,6}*216b
6-fold quotients : {12,6}*144d
8-fold quotients : {6,6}*108
9-fold quotients : {4,6}*96
12-fold quotients : {6,6}*72a
18-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
36-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
72-fold quotients : {2,3}*12, {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*1728l, {12,6}*1728b, {24,6}*1728c, {24,6}*1728e, {12,12}*1728p
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)
( 23, 29)( 24, 30)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 71)( 54, 72)( 55, 69)( 56, 70)
( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 73, 75)( 74, 76)( 77, 83)( 78, 84)
( 79, 81)( 80, 82)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,107)( 90,108)
( 91,105)( 92,106)( 93,103)( 94,104)( 95,101)( 96,102)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,135)(122,136)(123,133)(124,134)
(125,143)(126,144)(127,141)(128,142)(129,139)(130,140)(131,137)(132,138)
(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,171)(158,172)
(159,169)(160,170)(161,179)(162,180)(163,177)(164,178)(165,175)(166,176)
(167,173)(168,174)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,207)(194,208)(195,205)(196,206)(197,215)(198,216)(199,213)(200,214)
(201,211)(202,212)(203,209)(204,210);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 17)(  6, 19)(  7, 18)(  8, 20)
(  9, 21)( 10, 23)( 11, 22)( 12, 24)( 26, 27)( 30, 31)( 34, 35)( 37, 85)
( 38, 87)( 39, 86)( 40, 88)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 93)
( 46, 95)( 47, 94)( 48, 96)( 49, 73)( 50, 75)( 51, 74)( 52, 76)( 53, 77)
( 54, 79)( 55, 78)( 56, 80)( 57, 81)( 58, 83)( 59, 82)( 60, 84)( 61, 97)
( 62, 99)( 63, 98)( 64,100)( 65,101)( 66,103)( 67,102)( 68,104)( 69,105)
( 70,107)( 71,106)( 72,108)(109,121)(110,123)(111,122)(112,124)(113,125)
(114,127)(115,126)(116,128)(117,129)(118,131)(119,130)(120,132)(134,135)
(138,139)(142,143)(145,193)(146,195)(147,194)(148,196)(149,197)(150,199)
(151,198)(152,200)(153,201)(154,203)(155,202)(156,204)(157,181)(158,183)
(159,182)(160,184)(161,185)(162,187)(163,186)(164,188)(165,189)(166,191)
(167,190)(168,192)(169,205)(170,207)(171,206)(172,208)(173,209)(174,211)
(175,210)(176,212)(177,213)(178,215)(179,214)(180,216);;
s2 := (  1,181)(  2,184)(  3,183)(  4,182)(  5,189)(  6,192)(  7,191)(  8,190)
(  9,185)( 10,188)( 11,187)( 12,186)( 13,201)( 14,204)( 15,203)( 16,202)
( 17,197)( 18,200)( 19,199)( 20,198)( 21,193)( 22,196)( 23,195)( 24,194)
( 25,209)( 26,212)( 27,211)( 28,210)( 29,205)( 30,208)( 31,207)( 32,206)
( 33,213)( 34,216)( 35,215)( 36,214)( 37,145)( 38,148)( 39,147)( 40,146)
( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)( 48,150)
( 49,165)( 50,168)( 51,167)( 52,166)( 53,161)( 54,164)( 55,163)( 56,162)
( 57,157)( 58,160)( 59,159)( 60,158)( 61,173)( 62,176)( 63,175)( 64,174)
( 65,169)( 66,172)( 67,171)( 68,170)( 69,177)( 70,180)( 71,179)( 72,178)
( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)( 80,118)
( 81,113)( 82,116)( 83,115)( 84,114)( 85,129)( 86,132)( 87,131)( 88,130)
( 89,125)( 90,128)( 91,127)( 92,126)( 93,121)( 94,124)( 95,123)( 96,122)
( 97,137)( 98,140)( 99,139)(100,138)(101,133)(102,136)(103,135)(104,134)
(105,141)(106,144)(107,143)(108,142);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope