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# Polytope of Type {8,12}

Atlas Canonical Name : {8,12}*384f
if this polytope has a name.
Group : SmallGroup(384,17944)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 16, 96, 24
Order of s0s1s2 : 12
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,12,2} of size 768
Vertex Figure Of :
{2,8,12} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12}*192b, {8,6}*192c
4-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
8-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {4,3}*24, {2,6}*24
24-fold quotients : {2,4}*16
32-fold quotients : {2,3}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,24}*768k, {8,24}*768l, {8,12}*768s
3-fold covers : {8,36}*1152f, {24,12}*1152i, {24,12}*1152j
5-fold covers : {40,12}*1920e, {8,60}*1920f
Permutation Representation (GAP) :
```s0 := (  1,101)(  2,102)(  3,104)(  4,103)(  5, 98)(  6, 97)(  7, 99)(  8,100)
(  9,109)( 10,110)( 11,112)( 12,111)( 13,106)( 14,105)( 15,107)( 16,108)
( 17,117)( 18,118)( 19,120)( 20,119)( 21,114)( 22,113)( 23,115)( 24,116)
( 25,125)( 26,126)( 27,128)( 28,127)( 29,122)( 30,121)( 31,123)( 32,124)
( 33,133)( 34,134)( 35,136)( 36,135)( 37,130)( 38,129)( 39,131)( 40,132)
( 41,141)( 42,142)( 43,144)( 44,143)( 45,138)( 46,137)( 47,139)( 48,140)
( 49,150)( 50,149)( 51,151)( 52,152)( 53,145)( 54,146)( 55,148)( 56,147)
( 57,158)( 58,157)( 59,159)( 60,160)( 61,153)( 62,154)( 63,156)( 64,155)
( 65,166)( 66,165)( 67,167)( 68,168)( 69,161)( 70,162)( 71,164)( 72,163)
( 73,174)( 74,173)( 75,175)( 76,176)( 77,169)( 78,170)( 79,172)( 80,171)
( 81,182)( 82,181)( 83,183)( 84,184)( 85,177)( 86,178)( 87,180)( 88,179)
( 89,190)( 90,189)( 91,191)( 92,192)( 93,185)( 94,186)( 95,188)( 96,187);;
s1 := (  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)
( 14, 19)( 15, 24)( 16, 23)( 27, 30)( 28, 29)( 31, 32)( 33, 41)( 34, 42)
( 35, 46)( 36, 45)( 37, 44)( 38, 43)( 39, 48)( 40, 47)( 49, 73)( 50, 74)
( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 80)( 56, 79)( 57, 89)( 58, 90)
( 59, 94)( 60, 93)( 61, 92)( 62, 91)( 63, 96)( 64, 95)( 65, 81)( 66, 82)
( 67, 86)( 68, 85)( 69, 84)( 70, 83)( 71, 88)( 72, 87)( 99,102)(100,101)
(103,104)(105,113)(106,114)(107,118)(108,117)(109,116)(110,115)(111,120)
(112,119)(123,126)(124,125)(127,128)(129,137)(130,138)(131,142)(132,141)
(133,140)(134,139)(135,144)(136,143)(145,169)(146,170)(147,174)(148,173)
(149,172)(150,171)(151,176)(152,175)(153,185)(154,186)(155,190)(156,189)
(157,188)(158,187)(159,192)(160,191)(161,177)(162,178)(163,182)(164,181)
(165,180)(166,179)(167,184)(168,183);;
s2 := (  1, 65)(  2, 66)(  3, 71)(  4, 72)(  5, 70)(  6, 69)(  7, 67)(  8, 68)
(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)( 16, 60)
( 17, 49)( 18, 50)( 19, 55)( 20, 56)( 21, 54)( 22, 53)( 23, 51)( 24, 52)
( 25, 89)( 26, 90)( 27, 95)( 28, 96)( 29, 94)( 30, 93)( 31, 91)( 32, 92)
( 33, 81)( 34, 82)( 35, 87)( 36, 88)( 37, 86)( 38, 85)( 39, 83)( 40, 84)
( 41, 73)( 42, 74)( 43, 79)( 44, 80)( 45, 78)( 46, 77)( 47, 75)( 48, 76)
( 97,161)( 98,162)( 99,167)(100,168)(101,166)(102,165)(103,163)(104,164)
(105,153)(106,154)(107,159)(108,160)(109,158)(110,157)(111,155)(112,156)
(113,145)(114,146)(115,151)(116,152)(117,150)(118,149)(119,147)(120,148)
(121,185)(122,186)(123,191)(124,192)(125,190)(126,189)(127,187)(128,188)
(129,177)(130,178)(131,183)(132,184)(133,182)(134,181)(135,179)(136,180)
(137,169)(138,170)(139,175)(140,176)(141,174)(142,173)(143,171)(144,172);;
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*s1*s2*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope