Polytope of Type {3,8,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,4}*768c
Also Known As : {{3,8}6,{8,4|2}}. if this polytope has another name.
Group : SmallGroup(768,1087795)
Rank : 4
Schlafli Type : {3,8,4}
Number of vertices, edges, etc : 12, 48, 64, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
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 : {3,8,2}*384
4-fold quotients : {3,4,4}*192b
8-fold quotients : {3,4,2}*96
16-fold quotients : {3,2,4}*48, {3,4,2}*48
32-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
4 facets:
4 of 2-fold non-regular quotient of {3,8}*192
8 vertex figures:
4 of {8,4}*64a
4 of {4,4}*32
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
4 facets:
4 of 4-fold non-regular quotient of {3,8}*192
6 vertex figures:
2 of {8,4}*64a
4 of {2,4}*16
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
4 facets:
4 of 4-fold non-regular quotient of {3,8}*192
4 vertex figures:
2 of {8,4}*64a
2 of {4,4}*32
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 53, 54)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(101,102)(105,109)(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,134)(118,133)(119,135)(120,136)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)(147,148)(149,150)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)(168,184)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s1 := ( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 44)( 6, 42)( 7, 43)( 8, 41)( 9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)( 16, 48)( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)( 50, 83)( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)( 58, 86)( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)(108,133)(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)(119,123)(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)(150,186)(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)(158,191)(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)(174,175);;
s2 := ( 1,103)( 2,104)( 3,101)( 4,102)( 5, 99)( 6,100)( 7, 97)( 8, 98)( 9,109)( 10,110)( 11,111)( 12,112)( 13,105)( 14,106)( 15,107)( 16,108)( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)( 24,114)( 25,125)( 26,126)( 27,127)( 28,128)( 29,121)( 30,122)( 31,123)( 32,124)( 33,135)( 34,136)( 35,133)( 36,134)( 37,131)( 38,132)( 39,129)( 40,130)( 41,141)( 42,142)( 43,143)( 44,144)( 45,137)( 46,138)( 47,139)( 48,140)( 49,151)( 50,152)( 51,149)( 52,150)( 53,147)( 54,148)( 55,145)( 56,146)( 57,157)( 58,158)( 59,159)( 60,160)( 61,153)( 62,154)( 63,155)( 64,156)( 65,167)( 66,168)( 67,165)( 68,166)( 69,163)( 70,164)( 71,161)( 72,162)( 73,173)( 74,174)( 75,175)( 76,176)( 77,169)( 78,170)( 79,171)( 80,172)( 81,183)( 82,184)( 83,181)( 84,182)( 85,179)( 86,180)( 87,177)( 88,178)( 89,189)( 90,190)( 91,191)( 92,192)( 93,185)( 94,186)( 95,187)( 96,188);;
s3 := ( 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);;
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*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 3, 4)( 5, 6)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 53, 54)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(101,102)(105,109)(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,134)(118,133)(119,135)(120,136)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)(147,148)(149,150)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)(168,184)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);
s1 := Sym(192)!( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 44)( 6, 42)( 7, 43)( 8, 41)( 9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)( 16, 48)( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)( 50, 83)( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)( 58, 86)( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)(108,133)(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)(119,123)(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)(150,186)(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)(158,191)(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)(174,175);
s2 := Sym(192)!( 1,103)( 2,104)( 3,101)( 4,102)( 5, 99)( 6,100)( 7, 97)( 8, 98)( 9,109)( 10,110)( 11,111)( 12,112)( 13,105)( 14,106)( 15,107)( 16,108)( 17,119)( 18,120)( 19,117)( 20,118)( 21,115)( 22,116)( 23,113)( 24,114)( 25,125)( 26,126)( 27,127)( 28,128)( 29,121)( 30,122)( 31,123)( 32,124)( 33,135)( 34,136)( 35,133)( 36,134)( 37,131)( 38,132)( 39,129)( 40,130)( 41,141)( 42,142)( 43,143)( 44,144)( 45,137)( 46,138)( 47,139)( 48,140)( 49,151)( 50,152)( 51,149)( 52,150)( 53,147)( 54,148)( 55,145)( 56,146)( 57,157)( 58,158)( 59,159)( 60,160)( 61,153)( 62,154)( 63,155)( 64,156)( 65,167)( 66,168)( 67,165)( 68,166)( 69,163)( 70,164)( 71,161)( 72,162)( 73,173)( 74,174)( 75,175)( 76,176)( 77,169)( 78,170)( 79,171)( 80,172)( 81,183)( 82,184)( 83,181)( 84,182)( 85,179)( 86,180)( 87,177)( 88,178)( 89,189)( 90,190)( 91,191)( 92,192)( 93,185)( 94,186)( 95,187)( 96,188);
s3 := Sym(192)!( 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);
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*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope