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# Polytope of Type {12,8}

Atlas Canonical Name : {12,8}*384a
if this polytope has a name.
Group : SmallGroup(384,860)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 24, 96, 16
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,8,2} of size 768
Vertex Figure Of :
{2,12,8} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,4}*192a, {12,8}*192a, {12,8}*192b
3-fold quotients : {4,8}*128a
4-fold quotients : {12,4}*96a, {6,8}*96
6-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
8-fold quotients : {12,2}*48, {6,4}*48a
12-fold quotients : {4,4}*32, {2,8}*32
16-fold quotients : {6,2}*24
24-fold quotients : {2,4}*16, {4,2}*16
32-fold quotients : {3,2}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,8}*768a, {12,8}*768a, {24,8}*768c, {12,16}*768a, {12,16}*768b
3-fold covers : {36,8}*1152a, {12,24}*1152b, {12,24}*1152c
5-fold covers : {60,8}*1920a, {12,40}*1920a
Permutation Representation (GAP) :
```s0 := (  1, 49)(  2, 51)(  3, 50)(  4, 52)(  5, 54)(  6, 53)(  7, 55)(  8, 57)
(  9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)( 16, 64)
( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)( 24, 71)
( 25, 76)( 26, 78)( 27, 77)( 28, 73)( 29, 75)( 30, 74)( 31, 82)( 32, 84)
( 33, 83)( 34, 79)( 35, 81)( 36, 80)( 37, 88)( 38, 90)( 39, 89)( 40, 85)
( 41, 87)( 42, 86)( 43, 94)( 44, 96)( 45, 95)( 46, 91)( 47, 93)( 48, 92)
( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)(104,153)
(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)(112,160)
(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)(120,167)
(121,172)(122,174)(123,173)(124,169)(125,171)(126,170)(127,178)(128,180)
(129,179)(130,175)(131,177)(132,176)(133,184)(134,186)(135,185)(136,181)
(137,183)(138,182)(139,190)(140,192)(141,191)(142,187)(143,189)(144,188);;
s1 := (  1,  2)(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)
( 25, 32)( 26, 31)( 27, 33)( 28, 35)( 29, 34)( 30, 36)( 37, 44)( 38, 43)
( 39, 45)( 40, 47)( 41, 46)( 42, 48)( 49, 62)( 50, 61)( 51, 63)( 52, 65)
( 53, 64)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 71)( 59, 70)( 60, 72)
( 73, 92)( 74, 91)( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 86)( 80, 85)
( 81, 87)( 82, 89)( 83, 88)( 84, 90)( 97,122)( 98,121)( 99,123)(100,125)
(101,124)(102,126)(103,128)(104,127)(105,129)(106,131)(107,130)(108,132)
(109,134)(110,133)(111,135)(112,137)(113,136)(114,138)(115,140)(116,139)
(117,141)(118,143)(119,142)(120,144)(145,185)(146,184)(147,186)(148,182)
(149,181)(150,183)(151,191)(152,190)(153,192)(154,188)(155,187)(156,189)
(157,173)(158,172)(159,174)(160,170)(161,169)(162,171)(163,179)(164,178)
(165,180)(166,176)(167,175)(168,177);;
s2 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)(  8,104)
(  9,105)( 10,106)( 11,107)( 12,108)( 13,112)( 14,113)( 15,114)( 16,109)
( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)( 24,117)
( 25,127)( 26,128)( 27,129)( 28,130)( 29,131)( 30,132)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,142)( 38,143)( 39,144)( 40,139)
( 41,140)( 42,141)( 43,136)( 44,137)( 45,138)( 46,133)( 47,134)( 48,135)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)( 64,157)
( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)( 72,165)
( 73,175)( 74,176)( 75,177)( 76,178)( 77,179)( 78,180)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,190)( 86,191)( 87,192)( 88,187)
( 89,188)( 90,189)( 91,184)( 92,185)( 93,186)( 94,181)( 95,182)( 96,183);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope