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# Polytope of Type {8,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*768v
if this polytope has a name.
Group : SmallGroup(768,1087795)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 32, 192, 48
Order of s0s1s2 : 6
Order of s0s1s2s1 : 8
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 : {8,6}*384d
4-fold quotients : {8,6}*192a, {4,12}*192c
8-fold quotients : {4,6}*96
16-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
32-fold quotients : {4,3}*24, {2,6}*24
64-fold quotients : {2,3}*12
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,112)(  2,111)(  3,110)(  4,109)(  5,108)(  6,107)(  7,106)(  8,105)
(  9,104)( 10,103)( 11,102)( 12,101)( 13,100)( 14, 99)( 15, 98)( 16, 97)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,122)( 24,121)
( 25,120)( 26,119)( 27,118)( 28,117)( 29,116)( 30,115)( 31,114)( 32,113)
( 33,144)( 34,143)( 35,142)( 36,141)( 37,140)( 38,139)( 39,138)( 40,137)
( 41,136)( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)( 48,129)
( 49,160)( 50,159)( 51,158)( 52,157)( 53,156)( 54,155)( 55,154)( 56,153)
( 57,152)( 58,151)( 59,150)( 60,149)( 61,148)( 62,147)( 63,146)( 64,145)
( 65,176)( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,170)( 72,169)
( 73,168)( 74,167)( 75,166)( 76,165)( 77,164)( 78,163)( 79,162)( 80,161)
( 81,192)( 82,191)( 83,190)( 84,189)( 85,188)( 86,187)( 87,186)( 88,185)
( 89,184)( 90,183)( 91,182)( 92,181)( 93,180)( 94,179)( 95,178)( 96,177);;
s1 := (  3,  4)(  5,  6)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 53, 54)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)(100,147)
(101,150)(102,149)(103,151)(104,152)(105,157)(106,158)(107,160)(108,159)
(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)(116,179)
(117,182)(118,181)(119,183)(120,184)(121,189)(122,190)(123,192)(124,191)
(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)(132,163)
(133,166)(134,165)(135,167)(136,168)(137,173)(138,174)(139,176)(140,175)
(141,169)(142,170)(143,172)(144,171);;
s2 := (  1,144)(  2,142)(  3,143)(  4,141)(  5,133)(  6,135)(  7,134)(  8,136)
(  9,137)( 10,139)( 11,138)( 12,140)( 13,132)( 14,130)( 15,131)( 16,129)
( 17,128)( 18,126)( 19,127)( 20,125)( 21,117)( 22,119)( 23,118)( 24,120)
( 25,121)( 26,123)( 27,122)( 28,124)( 29,116)( 30,114)( 31,115)( 32,113)
( 33,112)( 34,110)( 35,111)( 36,109)( 37,101)( 38,103)( 39,102)( 40,104)
( 41,105)( 42,107)( 43,106)( 44,108)( 45,100)( 46, 98)( 47, 99)( 48, 97)
( 49,192)( 50,190)( 51,191)( 52,189)( 53,181)( 54,183)( 55,182)( 56,184)
( 57,185)( 58,187)( 59,186)( 60,188)( 61,180)( 62,178)( 63,179)( 64,177)
( 65,176)( 66,174)( 67,175)( 68,173)( 69,165)( 70,167)( 71,166)( 72,168)
( 73,169)( 74,171)( 75,170)( 76,172)( 77,164)( 78,162)( 79,163)( 80,161)
( 81,160)( 82,158)( 83,159)( 84,157)( 85,149)( 86,151)( 87,150)( 88,152)
( 89,153)( 90,155)( 91,154)( 92,156)( 93,148)( 94,146)( 95,147)( 96,145);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*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)!(  1,112)(  2,111)(  3,110)(  4,109)(  5,108)(  6,107)(  7,106)
(  8,105)(  9,104)( 10,103)( 11,102)( 12,101)( 13,100)( 14, 99)( 15, 98)
( 16, 97)( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,122)
( 24,121)( 25,120)( 26,119)( 27,118)( 28,117)( 29,116)( 30,115)( 31,114)
( 32,113)( 33,144)( 34,143)( 35,142)( 36,141)( 37,140)( 38,139)( 39,138)
( 40,137)( 41,136)( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)
( 48,129)( 49,160)( 50,159)( 51,158)( 52,157)( 53,156)( 54,155)( 55,154)
( 56,153)( 57,152)( 58,151)( 59,150)( 60,149)( 61,148)( 62,147)( 63,146)
( 64,145)( 65,176)( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,170)
( 72,169)( 73,168)( 74,167)( 75,166)( 76,165)( 77,164)( 78,163)( 79,162)
( 80,161)( 81,192)( 82,191)( 83,190)( 84,189)( 85,188)( 86,187)( 87,186)
( 88,185)( 89,184)( 90,183)( 91,182)( 92,181)( 93,180)( 94,179)( 95,178)
( 96,177);
s1 := Sym(192)!(  3,  4)(  5,  6)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 53, 54)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)
(100,147)(101,150)(102,149)(103,151)(104,152)(105,157)(106,158)(107,160)
(108,159)(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)
(116,179)(117,182)(118,181)(119,183)(120,184)(121,189)(122,190)(123,192)
(124,191)(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)
(132,163)(133,166)(134,165)(135,167)(136,168)(137,173)(138,174)(139,176)
(140,175)(141,169)(142,170)(143,172)(144,171);
s2 := Sym(192)!(  1,144)(  2,142)(  3,143)(  4,141)(  5,133)(  6,135)(  7,134)
(  8,136)(  9,137)( 10,139)( 11,138)( 12,140)( 13,132)( 14,130)( 15,131)
( 16,129)( 17,128)( 18,126)( 19,127)( 20,125)( 21,117)( 22,119)( 23,118)
( 24,120)( 25,121)( 26,123)( 27,122)( 28,124)( 29,116)( 30,114)( 31,115)
( 32,113)( 33,112)( 34,110)( 35,111)( 36,109)( 37,101)( 38,103)( 39,102)
( 40,104)( 41,105)( 42,107)( 43,106)( 44,108)( 45,100)( 46, 98)( 47, 99)
( 48, 97)( 49,192)( 50,190)( 51,191)( 52,189)( 53,181)( 54,183)( 55,182)
( 56,184)( 57,185)( 58,187)( 59,186)( 60,188)( 61,180)( 62,178)( 63,179)
( 64,177)( 65,176)( 66,174)( 67,175)( 68,173)( 69,165)( 70,167)( 71,166)
( 72,168)( 73,169)( 74,171)( 75,170)( 76,172)( 77,164)( 78,162)( 79,163)
( 80,161)( 81,160)( 82,158)( 83,159)( 84,157)( 85,149)( 86,151)( 87,150)
( 88,152)( 89,153)( 90,155)( 91,154)( 92,156)( 93,148)( 94,146)( 95,147)
( 96,145);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*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