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# Polytope of Type {8,12}

Atlas Canonical Name : {8,12}*384e
if this polytope has a name.
Group : SmallGroup(384,17922)
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
Petrial
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}*192b
4-fold quotients : {4,12}*96b, {4,12}*96c, {8,3}*96, {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}*768i, {8,24}*768j, {8,12}*768s
3-fold covers : {8,36}*1152e, {24,12}*1152k, {24,12}*1152l
5-fold covers : {40,12}*1920f, {8,60}*1920e
Permutation Representation (GAP) :
s0 := (  1, 53)(  2, 54)(  3, 56)(  4, 55)(  5, 50)(  6, 49)(  7, 51)(  8, 52)
(  9, 61)( 10, 62)( 11, 64)( 12, 63)( 13, 58)( 14, 57)( 15, 59)( 16, 60)
( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 66)( 22, 65)( 23, 67)( 24, 68)
( 25, 77)( 26, 78)( 27, 80)( 28, 79)( 29, 74)( 30, 73)( 31, 75)( 32, 76)
( 33, 85)( 34, 86)( 35, 88)( 36, 87)( 37, 82)( 38, 81)( 39, 83)( 40, 84)
( 41, 93)( 42, 94)( 43, 96)( 44, 95)( 45, 90)( 46, 89)( 47, 91)( 48, 92)
( 97,149)( 98,150)( 99,152)(100,151)(101,146)(102,145)(103,147)(104,148)
(105,157)(106,158)(107,160)(108,159)(109,154)(110,153)(111,155)(112,156)
(113,165)(114,166)(115,168)(116,167)(117,162)(118,161)(119,163)(120,164)
(121,173)(122,174)(123,176)(124,175)(125,170)(126,169)(127,171)(128,172)
(129,181)(130,182)(131,184)(132,183)(133,178)(134,177)(135,179)(136,180)
(137,189)(138,190)(139,192)(140,191)(141,186)(142,185)(143,187)(144,188);;
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, 50)( 51, 53)
( 52, 54)( 57, 66)( 58, 65)( 59, 69)( 60, 70)( 61, 67)( 62, 68)( 63, 71)
( 64, 72)( 73, 74)( 75, 77)( 76, 78)( 81, 90)( 82, 89)( 83, 93)( 84, 94)
( 85, 91)( 86, 92)( 87, 95)( 88, 96)( 97,121)( 98,122)( 99,126)(100,125)
(101,124)(102,123)(103,128)(104,127)(105,137)(106,138)(107,142)(108,141)
(109,140)(110,139)(111,144)(112,143)(113,129)(114,130)(115,134)(116,133)
(117,132)(118,131)(119,136)(120,135)(145,170)(146,169)(147,173)(148,174)
(149,171)(150,172)(151,175)(152,176)(153,186)(154,185)(155,189)(156,190)
(157,187)(158,188)(159,191)(160,192)(161,178)(162,177)(163,181)(164,182)
(165,179)(166,180)(167,183)(168,184);;
s2 := (  1,113)(  2,114)(  3,119)(  4,120)(  5,118)(  6,117)(  7,115)(  8,116)
(  9,105)( 10,106)( 11,111)( 12,112)( 13,110)( 14,109)( 15,107)( 16,108)
( 17, 97)( 18, 98)( 19,103)( 20,104)( 21,102)( 22,101)( 23, 99)( 24,100)
( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)( 32,140)
( 33,129)( 34,130)( 35,135)( 36,136)( 37,134)( 38,133)( 39,131)( 40,132)
( 41,121)( 42,122)( 43,127)( 44,128)( 45,126)( 46,125)( 47,123)( 48,124)
( 49,162)( 50,161)( 51,168)( 52,167)( 53,165)( 54,166)( 55,164)( 56,163)
( 57,154)( 58,153)( 59,160)( 60,159)( 61,157)( 62,158)( 63,156)( 64,155)
( 65,146)( 66,145)( 67,152)( 68,151)( 69,149)( 70,150)( 71,148)( 72,147)
( 73,186)( 74,185)( 75,192)( 76,191)( 77,189)( 78,190)( 79,188)( 80,187)
( 81,178)( 82,177)( 83,184)( 84,183)( 85,181)( 86,182)( 87,180)( 88,179)
( 89,170)( 90,169)( 91,176)( 92,175)( 93,173)( 94,174)( 95,172)( 96,171);;
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*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

References : None.
to this polytope