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# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*768e
if this polytope has a name.
Group : SmallGroup(768,1086324)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 64, 192, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,6}*384e
4-fold quotients : {3,6}*192, {12,6}*192b
8-fold quotients : {12,3}*96, {6,6}*96
16-fold quotients : {3,6}*48, {6,3}*48
32-fold quotients : {3,3}*24
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  6)(  9, 15)( 10, 16)( 11, 14)( 12, 13)( 17, 28)( 18, 27)
( 19, 25)( 20, 26)( 21, 31)( 22, 32)( 23, 30)( 24, 29)( 33, 65)( 34, 66)
( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 71)( 40, 72)( 41, 79)( 42, 80)
( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 73)( 48, 74)( 49, 92)( 50, 91)
( 51, 89)( 52, 90)( 53, 95)( 54, 96)( 55, 94)( 56, 93)( 57, 83)( 58, 84)
( 59, 82)( 60, 81)( 61, 88)( 62, 87)( 63, 85)( 64, 86)( 97,103)( 98,104)
( 99,102)(100,101)(107,108)(109,110)(113,126)(114,125)(115,127)(116,128)
(117,121)(118,122)(119,124)(120,123)(129,167)(130,168)(131,166)(132,165)
(133,164)(134,163)(135,161)(136,162)(137,169)(138,170)(139,172)(140,171)
(141,174)(142,173)(143,175)(144,176)(145,190)(146,189)(147,191)(148,192)
(149,185)(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)
(157,178)(158,177)(159,179)(160,180);;
s1 := (  1,167)(  2,165)(  3,168)(  4,166)(  5,162)(  6,164)(  7,161)(  8,163)
(  9,178)( 10,180)( 11,177)( 12,179)( 13,183)( 14,181)( 15,184)( 16,182)
( 17,171)( 18,169)( 19,172)( 20,170)( 21,174)( 22,176)( 23,173)( 24,175)
( 25,188)( 26,186)( 27,187)( 28,185)( 29,189)( 30,191)( 31,190)( 32,192)
( 33,135)( 34,133)( 35,136)( 36,134)( 37,130)( 38,132)( 39,129)( 40,131)
( 41,146)( 42,148)( 43,145)( 44,147)( 45,151)( 46,149)( 47,152)( 48,150)
( 49,139)( 50,137)( 51,140)( 52,138)( 53,142)( 54,144)( 55,141)( 56,143)
( 57,156)( 58,154)( 59,155)( 60,153)( 61,157)( 62,159)( 63,158)( 64,160)
( 65,103)( 66,101)( 67,104)( 68,102)( 69, 98)( 70,100)( 71, 97)( 72, 99)
( 73,114)( 74,116)( 75,113)( 76,115)( 77,119)( 78,117)( 79,120)( 80,118)
( 81,107)( 82,105)( 83,108)( 84,106)( 85,110)( 86,112)( 87,109)( 88,111)
( 89,124)( 90,122)( 91,123)( 92,121)( 93,125)( 94,127)( 95,126)( 96,128);;
s2 := (  1,106)(  2,105)(  3,107)(  4,108)(  5,109)(  6,110)(  7,112)(  8,111)
(  9, 98)( 10, 97)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,104)( 16,103)
( 17,119)( 18,120)( 19,118)( 20,117)( 21,116)( 22,115)( 23,113)( 24,114)
( 25,122)( 26,121)( 27,123)( 28,124)( 29,125)( 30,126)( 31,128)( 32,127)
( 33,170)( 34,169)( 35,171)( 36,172)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,162)( 42,161)( 43,163)( 44,164)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,183)( 50,184)( 51,182)( 52,181)( 53,180)( 54,179)( 55,177)( 56,178)
( 57,186)( 58,185)( 59,187)( 60,188)( 61,189)( 62,190)( 63,192)( 64,191)
( 65,138)( 66,137)( 67,139)( 68,140)( 69,141)( 70,142)( 71,144)( 72,143)
( 73,130)( 74,129)( 75,131)( 76,132)( 77,133)( 78,134)( 79,136)( 80,135)
( 81,151)( 82,152)( 83,150)( 84,149)( 85,148)( 86,147)( 87,145)( 88,146)
( 89,154)( 90,153)( 91,155)( 92,156)( 93,157)( 94,158)( 95,160)( 96,159);;
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*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope