Polytope of Type {4,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1087581)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 4, 48, 48, 16
Order of s0s1s2s3 : 12
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 : {2,6,4}*384a
4-fold quotients : {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.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2> of order 2.
8 facets:
8 of {4,6}*48a
4 vertex figures:
4 of 2-fold non-regular quotient of {6,4}*192a
P/N, where N=<s1*s2*s3*s2*s1*s2*s3*s2*s3> of order 2.
8 facets:
8 of {4,6}*48a
4 vertex figures:
4 of 2-fold non-regular quotient of {6,4}*192a
P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2, s1*s2*s3*s2*s1*s2*s3*s2*s3> of order 4.
4 facets:
4 of {4,6}*48a
4 vertex figures:
4 of 4-fold non-regular quotient of {6,4}*192a
P/N, where N=<s1*s2*s3*s2*s1*s3, s2*s1*s2*s3*s2*s1*s3*s2> of order 4.
4 facets:
4 of {4,6}*48a
4 vertex figures:
4 of 4-fold non-regular quotient of {6,4}*192a
Permutation Representation (GAP) :
s0 := ( 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, 97)( 2, 98)( 3,100)( 4, 99)( 5,102)( 6,101)( 7,103)( 8,104)( 9,112)( 10,111)( 11,109)( 12,110)( 13,107)( 14,108)( 15,106)( 16,105)( 17,129)( 18,130)( 19,132)( 20,131)( 21,134)( 22,133)( 23,135)( 24,136)( 25,144)( 26,143)( 27,141)( 28,142)( 29,139)( 30,140)( 31,138)( 32,137)( 33,113)( 34,114)( 35,116)( 36,115)( 37,118)( 38,117)( 39,119)( 40,120)( 41,128)( 42,127)( 43,125)( 44,126)( 45,123)( 46,124)( 47,122)( 48,121)( 49,145)( 50,146)( 51,148)( 52,147)( 53,150)( 54,149)( 55,151)( 56,152)( 57,160)( 58,159)( 59,157)( 60,158)( 61,155)( 62,156)( 63,154)( 64,153)( 65,177)( 66,178)( 67,180)( 68,179)( 69,182)( 70,181)( 71,183)( 72,184)( 73,192)( 74,191)( 75,189)( 76,190)( 77,187)( 78,188)( 79,186)( 80,185)( 81,161)( 82,162)( 83,164)( 84,163)( 85,166)( 86,165)( 87,167)( 88,168)( 89,176)( 90,175)( 91,173)( 92,174)( 93,171)( 94,172)( 95,170)( 96,169);;
s2 := ( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)( 8, 44)( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)( 16, 48)( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)( 50, 83)( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)(108,136)(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)(119,122)(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)(158,191)(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)(174,175);;
s3 := ( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 15)( 10, 16)( 11, 13)( 12, 14)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)( 60, 62)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 79)( 74, 80)( 75, 77)( 76, 78)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 95)( 90, 96)( 91, 93)( 92, 94)( 97,103)( 98,104)( 99,101)(100,102)(105,111)(106,112)(107,109)(108,110)(113,119)(114,120)(115,117)(116,118)(121,127)(122,128)(123,125)(124,126)(129,135)(130,136)(131,133)(132,134)(137,143)(138,144)(139,141)(140,142)(145,151)(146,152)(147,149)(148,150)(153,159)(154,160)(155,157)(156,158)(161,167)(162,168)(163,165)(164,166)(169,175)(170,176)(171,173)(172,174)(177,183)(178,184)(179,181)(180,182)(185,191)(186,192)(187,189)(188,190);;
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*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(192)!( 1, 97)( 2, 98)( 3,100)( 4, 99)( 5,102)( 6,101)( 7,103)( 8,104)( 9,112)( 10,111)( 11,109)( 12,110)( 13,107)( 14,108)( 15,106)( 16,105)( 17,129)( 18,130)( 19,132)( 20,131)( 21,134)( 22,133)( 23,135)( 24,136)( 25,144)( 26,143)( 27,141)( 28,142)( 29,139)( 30,140)( 31,138)( 32,137)( 33,113)( 34,114)( 35,116)( 36,115)( 37,118)( 38,117)( 39,119)( 40,120)( 41,128)( 42,127)( 43,125)( 44,126)( 45,123)( 46,124)( 47,122)( 48,121)( 49,145)( 50,146)( 51,148)( 52,147)( 53,150)( 54,149)( 55,151)( 56,152)( 57,160)( 58,159)( 59,157)( 60,158)( 61,155)( 62,156)( 63,154)( 64,153)( 65,177)( 66,178)( 67,180)( 68,179)( 69,182)( 70,181)( 71,183)( 72,184)( 73,192)( 74,191)( 75,189)( 76,190)( 77,187)( 78,188)( 79,186)( 80,185)( 81,161)( 82,162)( 83,164)( 84,163)( 85,166)( 86,165)( 87,167)( 88,168)( 89,176)( 90,175)( 91,173)( 92,174)( 93,171)( 94,172)( 95,170)( 96,169);
s2 := Sym(192)!( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)( 8, 44)( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)( 16, 48)( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)( 50, 83)( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)(108,136)(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)(119,122)(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)(158,191)(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)(174,175);
s3 := Sym(192)!( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 15)( 10, 16)( 11, 13)( 12, 14)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)( 60, 62)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 79)( 74, 80)( 75, 77)( 76, 78)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 95)( 90, 96)( 91, 93)( 92, 94)( 97,103)( 98,104)( 99,101)(100,102)(105,111)(106,112)(107,109)(108,110)(113,119)(114,120)(115,117)(116,118)(121,127)(122,128)(123,125)(124,126)(129,135)(130,136)(131,133)(132,134)(137,143)(138,144)(139,141)(140,142)(145,151)(146,152)(147,149)(148,150)(153,159)(154,160)(155,157)(156,158)(161,167)(162,168)(163,165)(164,166)(169,175)(170,176)(171,173)(172,174)(177,183)(178,184)(179,181)(180,182)(185,191)(186,192)(187,189)(188,190);
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*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 >;
References : None.
to this polytope