Polytope of Type {8,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1087755)
Rank : 4
Schlafli Type : {8,12,4}
Number of vertices, edges, etc : 8, 48, 24, 4
Order of s0s1s2s3 : 24
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 : {4,12,4}*384c, {8,6,4}*384b
4-fold quotients : {2,12,4}*192c, {4,6,4}*192b
8-fold quotients : {2,6,4}*96c
16-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)
( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 73, 85)( 74, 86)( 75, 87)( 76, 88)
( 77, 89)( 78, 90)( 79, 91)( 80, 92)( 81, 93)( 82, 94)( 83, 95)( 84, 96)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)
(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192);;
s1 := ( 1,121)( 2,122)( 3,124)( 4,123)( 5,129)( 6,130)( 7,132)( 8,131)
( 9,125)( 10,126)( 11,128)( 12,127)( 13,133)( 14,134)( 15,136)( 16,135)
( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)( 24,139)
( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)( 32,107)
( 33,101)( 34,102)( 35,104)( 36,103)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)
( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)( 56,191)
( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)( 64,171)
( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)( 72,175)
( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)( 80,167)
( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)( 88,147)
( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)( 96,151);;
s2 := ( 1, 9)( 2, 11)( 3, 10)( 4, 12)( 6, 7)( 13, 21)( 14, 23)( 15, 22)
( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)( 37, 45)
( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)( 52, 60)
( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)( 74, 83)
( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)( 90, 91)
( 97,129)( 98,131)( 99,130)(100,132)(101,125)(102,127)(103,126)(104,128)
(105,121)(106,123)(107,122)(108,124)(109,141)(110,143)(111,142)(112,144)
(113,137)(114,139)(115,138)(116,140)(117,133)(118,135)(119,134)(120,136)
(145,177)(146,179)(147,178)(148,180)(149,173)(150,175)(151,174)(152,176)
(153,169)(154,171)(155,170)(156,172)(157,189)(158,191)(159,190)(160,192)
(161,185)(162,187)(163,186)(164,188)(165,181)(166,183)(167,182)(168,184);;
s3 := ( 1, 26)( 2, 25)( 3, 28)( 4, 27)( 5, 30)( 6, 29)( 7, 32)( 8, 31)
( 9, 34)( 10, 33)( 11, 36)( 12, 35)( 13, 38)( 14, 37)( 15, 40)( 16, 39)
( 17, 42)( 18, 41)( 19, 44)( 20, 43)( 21, 46)( 22, 45)( 23, 48)( 24, 47)
( 49, 74)( 50, 73)( 51, 76)( 52, 75)( 53, 78)( 54, 77)( 55, 80)( 56, 79)
( 57, 82)( 58, 81)( 59, 84)( 60, 83)( 61, 86)( 62, 85)( 63, 88)( 64, 87)
( 65, 90)( 66, 89)( 67, 92)( 68, 91)( 69, 94)( 70, 93)( 71, 96)( 72, 95)
( 97,122)( 98,121)( 99,124)(100,123)(101,126)(102,125)(103,128)(104,127)
(105,130)(106,129)(107,132)(108,131)(109,134)(110,133)(111,136)(112,135)
(113,138)(114,137)(115,140)(116,139)(117,142)(118,141)(119,144)(120,143)
(145,170)(146,169)(147,172)(148,171)(149,174)(150,173)(151,176)(152,175)
(153,178)(154,177)(155,180)(156,179)(157,182)(158,181)(159,184)(160,183)
(161,186)(162,185)(163,188)(164,187)(165,190)(166,189)(167,192)(168,191);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*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)!( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 65)( 54, 66)( 55, 67)
( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 73, 85)( 74, 86)( 75, 87)
( 76, 88)( 77, 89)( 78, 90)( 79, 91)( 80, 92)( 81, 93)( 82, 94)( 83, 95)
( 84, 96)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)
(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)
(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)
(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)
(144,192);
s1 := Sym(192)!( 1,121)( 2,122)( 3,124)( 4,123)( 5,129)( 6,130)( 7,132)
( 8,131)( 9,125)( 10,126)( 11,128)( 12,127)( 13,133)( 14,134)( 15,136)
( 16,135)( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)
( 24,139)( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)
( 32,107)( 33,101)( 34,102)( 35,104)( 36,103)( 37,109)( 38,110)( 39,112)
( 40,111)( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)
( 48,115)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)
( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)
( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)
( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)
( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)
( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)
( 96,151);
s2 := Sym(192)!( 1, 9)( 2, 11)( 3, 10)( 4, 12)( 6, 7)( 13, 21)( 14, 23)
( 15, 22)( 16, 24)( 18, 19)( 25, 33)( 26, 35)( 27, 34)( 28, 36)( 30, 31)
( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 42, 43)( 49, 57)( 50, 59)( 51, 58)
( 52, 60)( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)( 73, 81)
( 74, 83)( 75, 82)( 76, 84)( 78, 79)( 85, 93)( 86, 95)( 87, 94)( 88, 96)
( 90, 91)( 97,129)( 98,131)( 99,130)(100,132)(101,125)(102,127)(103,126)
(104,128)(105,121)(106,123)(107,122)(108,124)(109,141)(110,143)(111,142)
(112,144)(113,137)(114,139)(115,138)(116,140)(117,133)(118,135)(119,134)
(120,136)(145,177)(146,179)(147,178)(148,180)(149,173)(150,175)(151,174)
(152,176)(153,169)(154,171)(155,170)(156,172)(157,189)(158,191)(159,190)
(160,192)(161,185)(162,187)(163,186)(164,188)(165,181)(166,183)(167,182)
(168,184);
s3 := Sym(192)!( 1, 26)( 2, 25)( 3, 28)( 4, 27)( 5, 30)( 6, 29)( 7, 32)
( 8, 31)( 9, 34)( 10, 33)( 11, 36)( 12, 35)( 13, 38)( 14, 37)( 15, 40)
( 16, 39)( 17, 42)( 18, 41)( 19, 44)( 20, 43)( 21, 46)( 22, 45)( 23, 48)
( 24, 47)( 49, 74)( 50, 73)( 51, 76)( 52, 75)( 53, 78)( 54, 77)( 55, 80)
( 56, 79)( 57, 82)( 58, 81)( 59, 84)( 60, 83)( 61, 86)( 62, 85)( 63, 88)
( 64, 87)( 65, 90)( 66, 89)( 67, 92)( 68, 91)( 69, 94)( 70, 93)( 71, 96)
( 72, 95)( 97,122)( 98,121)( 99,124)(100,123)(101,126)(102,125)(103,128)
(104,127)(105,130)(106,129)(107,132)(108,131)(109,134)(110,133)(111,136)
(112,135)(113,138)(114,137)(115,140)(116,139)(117,142)(118,141)(119,144)
(120,143)(145,170)(146,169)(147,172)(148,171)(149,174)(150,173)(151,176)
(152,175)(153,178)(154,177)(155,180)(156,179)(157,182)(158,181)(159,184)
(160,183)(161,186)(162,185)(163,188)(164,187)(165,190)(166,189)(167,192)
(168,191);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope