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# Polytope of Type {8,6,4}

Atlas Canonical Name : {8,6,4}*768g
if this polytope has a name.
Group : SmallGroup(768,1090200)
Rank : 4
Schlafli Type : {8,6,4}
Number of vertices, edges, etc : 16, 48, 24, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
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 : {4,6,4}*384d
4-fold quotients : {4,3,4}*192b, {4,6,4}*192d, {4,6,4}*192f
8-fold quotients : {2,6,4}*96c, {4,3,4}*96
16-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,104)(  2,103)(  3,102)(  4,101)(  5, 99)(  6,100)(  7, 97)(  8, 98)
(  9,112)( 10,111)( 11,110)( 12,109)( 13,107)( 14,108)( 15,105)( 16,106)
( 17,120)( 18,119)( 19,118)( 20,117)( 21,115)( 22,116)( 23,113)( 24,114)
( 25,128)( 26,127)( 27,126)( 28,125)( 29,123)( 30,124)( 31,121)( 32,122)
( 33,136)( 34,135)( 35,134)( 36,133)( 37,131)( 38,132)( 39,129)( 40,130)
( 41,144)( 42,143)( 43,142)( 44,141)( 45,139)( 46,140)( 47,137)( 48,138)
( 49,152)( 50,151)( 51,150)( 52,149)( 53,147)( 54,148)( 55,145)( 56,146)
( 57,160)( 58,159)( 59,158)( 60,157)( 61,155)( 62,156)( 63,153)( 64,154)
( 65,168)( 66,167)( 67,166)( 68,165)( 69,163)( 70,164)( 71,161)( 72,162)
( 73,176)( 74,175)( 75,174)( 76,173)( 77,171)( 78,172)( 79,169)( 80,170)
( 81,184)( 82,183)( 83,182)( 84,181)( 85,179)( 86,180)( 87,177)( 88,178)
( 89,192)( 90,191)( 91,190)( 92,189)( 93,187)( 94,188)( 95,185)( 96,186);;
s1 := (  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 17, 25)( 18, 26)
( 19, 28)( 20, 27)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)
( 35, 68)( 36, 67)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)
( 43, 76)( 44, 75)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)
( 51, 92)( 52, 91)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)
( 59, 84)( 60, 83)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 99,100)(101,103)
(102,104)(107,108)(109,111)(110,112)(113,121)(114,122)(115,124)(116,123)
(117,127)(118,128)(119,125)(120,126)(129,161)(130,162)(131,164)(132,163)
(133,167)(134,168)(135,165)(136,166)(137,169)(138,170)(139,172)(140,171)
(141,175)(142,176)(143,173)(144,174)(145,185)(146,186)(147,188)(148,187)
(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,180)(156,179)
(157,183)(158,184)(159,181)(160,182);;
s2 := (  1, 65)(  2, 66)(  3, 70)(  4, 69)(  5, 68)(  6, 67)(  7, 72)(  8, 71)
(  9, 81)( 10, 82)( 11, 86)( 12, 85)( 13, 84)( 14, 83)( 15, 88)( 16, 87)
( 17, 73)( 18, 74)( 19, 78)( 20, 77)( 21, 76)( 22, 75)( 23, 80)( 24, 79)
( 25, 89)( 26, 90)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)( 32, 95)
( 35, 38)( 36, 37)( 39, 40)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 56)( 48, 55)( 59, 62)( 60, 61)( 63, 64)( 97,162)( 98,161)
( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,178)(106,177)
(107,181)(108,182)(109,179)(110,180)(111,183)(112,184)(113,170)(114,169)
(115,173)(116,174)(117,171)(118,172)(119,175)(120,176)(121,186)(122,185)
(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(129,130)(131,133)
(132,134)(137,146)(138,145)(139,149)(140,150)(141,147)(142,148)(143,151)
(144,152)(153,154)(155,157)(156,158);;
s3 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)
(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)
(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)
(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(192)!(  1,104)(  2,103)(  3,102)(  4,101)(  5, 99)(  6,100)(  7, 97)
(  8, 98)(  9,112)( 10,111)( 11,110)( 12,109)( 13,107)( 14,108)( 15,105)
( 16,106)( 17,120)( 18,119)( 19,118)( 20,117)( 21,115)( 22,116)( 23,113)
( 24,114)( 25,128)( 26,127)( 27,126)( 28,125)( 29,123)( 30,124)( 31,121)
( 32,122)( 33,136)( 34,135)( 35,134)( 36,133)( 37,131)( 38,132)( 39,129)
( 40,130)( 41,144)( 42,143)( 43,142)( 44,141)( 45,139)( 46,140)( 47,137)
( 48,138)( 49,152)( 50,151)( 51,150)( 52,149)( 53,147)( 54,148)( 55,145)
( 56,146)( 57,160)( 58,159)( 59,158)( 60,157)( 61,155)( 62,156)( 63,153)
( 64,154)( 65,168)( 66,167)( 67,166)( 68,165)( 69,163)( 70,164)( 71,161)
( 72,162)( 73,176)( 74,175)( 75,174)( 76,173)( 77,171)( 78,172)( 79,169)
( 80,170)( 81,184)( 82,183)( 83,182)( 84,181)( 85,179)( 86,180)( 87,177)
( 88,178)( 89,192)( 90,191)( 91,190)( 92,189)( 93,187)( 94,188)( 95,185)
( 96,186);
s1 := Sym(192)!(  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 17, 25)
( 18, 26)( 19, 28)( 20, 27)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)
( 34, 66)( 35, 68)( 36, 67)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)
( 42, 74)( 43, 76)( 44, 75)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)
( 50, 90)( 51, 92)( 52, 91)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)
( 58, 82)( 59, 84)( 60, 83)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 99,100)
(101,103)(102,104)(107,108)(109,111)(110,112)(113,121)(114,122)(115,124)
(116,123)(117,127)(118,128)(119,125)(120,126)(129,161)(130,162)(131,164)
(132,163)(133,167)(134,168)(135,165)(136,166)(137,169)(138,170)(139,172)
(140,171)(141,175)(142,176)(143,173)(144,174)(145,185)(146,186)(147,188)
(148,187)(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,180)
(156,179)(157,183)(158,184)(159,181)(160,182);
s2 := Sym(192)!(  1, 65)(  2, 66)(  3, 70)(  4, 69)(  5, 68)(  6, 67)(  7, 72)
(  8, 71)(  9, 81)( 10, 82)( 11, 86)( 12, 85)( 13, 84)( 14, 83)( 15, 88)
( 16, 87)( 17, 73)( 18, 74)( 19, 78)( 20, 77)( 21, 76)( 22, 75)( 23, 80)
( 24, 79)( 25, 89)( 26, 90)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)
( 32, 95)( 35, 38)( 36, 37)( 39, 40)( 41, 49)( 42, 50)( 43, 54)( 44, 53)
( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 59, 62)( 60, 61)( 63, 64)( 97,162)
( 98,161)( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,178)
(106,177)(107,181)(108,182)(109,179)(110,180)(111,183)(112,184)(113,170)
(114,169)(115,173)(116,174)(117,171)(118,172)(119,175)(120,176)(121,186)
(122,185)(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(129,130)
(131,133)(132,134)(137,146)(138,145)(139,149)(140,150)(141,147)(142,148)
(143,151)(144,152)(153,154)(155,157)(156,158);
s3 := Sym(192)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)
(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)
(152,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)
(168,176)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)
(184,192);
poly := sub<Sym(192)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope