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# Polytope of Type {12,8}

Atlas Canonical Name : {12,8}*768u
if this polytope has a name.
Group : SmallGroup(768,1087755)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 48, 192, 32
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
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 : {12,4}*384d, {6,8}*384g
4-fold quotients : {12,8}*192a, {12,4}*192b, {6,4}*192b, {12,4}*192c
8-fold quotients : {12,4}*96a, {6,8}*96, {12,4}*96b, {12,4}*96c, {6,4}*96
12-fold quotients : {4,8}*64a
16-fold quotients : {12,2}*48, {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
24-fold quotients : {4,4}*32, {2,8}*32
32-fold quotients : {3,4}*24, {6,2}*24
48-fold quotients : {2,4}*16, {4,2}*16
64-fold quotients : {3,2}*12
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)( 77, 81)
( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 97,121)( 98,122)( 99,124)(100,123)(101,129)(102,130)(103,132)(104,131)
(105,125)(106,126)(107,128)(108,127)(109,133)(110,134)(111,136)(112,135)
(113,141)(114,142)(115,144)(116,143)(117,137)(118,138)(119,140)(120,139)
(145,169)(146,170)(147,172)(148,171)(149,177)(150,178)(151,180)(152,179)
(153,173)(154,174)(155,176)(156,175)(157,181)(158,182)(159,184)(160,183)
(161,189)(162,190)(163,192)(164,191)(165,185)(166,186)(167,188)(168,187);;
s1 := (  1,129)(  2,131)(  3,130)(  4,132)(  5,125)(  6,127)(  7,126)(  8,128)
(  9,121)( 10,123)( 11,122)( 12,124)( 13,141)( 14,143)( 15,142)( 16,144)
( 17,137)( 18,139)( 19,138)( 20,140)( 21,133)( 22,135)( 23,134)( 24,136)
( 25,105)( 26,107)( 27,106)( 28,108)( 29,101)( 30,103)( 31,102)( 32,104)
( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,117)( 38,119)( 39,118)( 40,120)
( 41,113)( 42,115)( 43,114)( 44,116)( 45,109)( 46,111)( 47,110)( 48,112)
( 49,189)( 50,191)( 51,190)( 52,192)( 53,185)( 54,187)( 55,186)( 56,188)
( 57,181)( 58,183)( 59,182)( 60,184)( 61,177)( 62,179)( 63,178)( 64,180)
( 65,173)( 66,175)( 67,174)( 68,176)( 69,169)( 70,171)( 71,170)( 72,172)
( 73,165)( 74,167)( 75,166)( 76,168)( 77,161)( 78,163)( 79,162)( 80,164)
( 81,157)( 82,159)( 83,158)( 84,160)( 85,153)( 86,155)( 87,154)( 88,156)
( 89,149)( 90,151)( 91,150)( 92,152)( 93,145)( 94,147)( 95,146)( 96,148);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 62)( 50, 61)( 51, 64)( 52, 63)( 53, 66)( 54, 65)( 55, 68)( 56, 67)
( 57, 70)( 58, 69)( 59, 72)( 60, 71)( 73, 86)( 74, 85)( 75, 88)( 76, 87)
( 77, 90)( 78, 89)( 79, 92)( 80, 91)( 81, 94)( 82, 93)( 83, 96)( 84, 95)
( 97,146)( 98,145)( 99,148)(100,147)(101,150)(102,149)(103,152)(104,151)
(105,154)(106,153)(107,156)(108,155)(109,158)(110,157)(111,160)(112,159)
(113,162)(114,161)(115,164)(116,163)(117,166)(118,165)(119,168)(120,167)
(121,170)(122,169)(123,172)(124,171)(125,174)(126,173)(127,176)(128,175)
(129,178)(130,177)(131,180)(132,179)(133,182)(134,181)(135,184)(136,183)
(137,186)(138,185)(139,188)(140,187)(141,190)(142,189)(143,192)(144,191);;
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*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*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope