Polytope of Type {6,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,6}*1152b
if this polytope has a name.
Group : SmallGroup(1152,157559)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 24, 48, 48, 6
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 : {6,4,6}*576b
3-fold quotients : {6,4,2}*384b
4-fold quotients : {6,4,6}*288, {3,4,6}*288
6-fold quotients : {6,4,2}*192
8-fold quotients : {6,2,6}*144
12-fold quotients : {2,4,6}*96a, {6,4,2}*96a, {3,4,2}*96, {6,4,2}*96b, {6,4,2}*96c
16-fold quotients : {3,2,6}*72, {6,2,3}*72
24-fold quotients : {3,4,2}*48, {2,2,6}*48, {6,2,2}*48
32-fold quotients : {3,2,3}*36
36-fold quotients : {2,4,2}*32
48-fold quotients : {2,2,3}*24, {3,2,2}*24
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2.
6 facets:
6 of 2-fold non-regular quotient of {6,4}*192b
16 vertex figures:
8 of {4,6}*48a
8 of {2,6}*24
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
6 facets:
6 of 2-fold non-regular quotient of {6,4}*192b
12 vertex figures:
12 of {4,6}*48a
P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
6 facets:
6 of 2-fold non-regular quotient of {6,4}*192b
12 vertex figures:
12 of {4,6}*48a
P/N, where N=<s0*s1*s0*s1> of order 3.
6 facets:
6 of 3-fold non-regular quotient of {6,4}*192b
8 vertex figures:
8 of {4,6}*48a
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
6 facets:
6 of 4-fold non-regular quotient of {6,4}*192b
10 vertex figures:
8 of {2,6}*24
2 of {4,6}*48a
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
6 facets:
6 of 4-fold non-regular quotient of {6,4}*192b
8 vertex figures:
4 of {4,6}*48a
4 of {2,6}*24
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
6 facets:
6 of 4-fold non-regular quotient of {6,4}*192b
6 vertex figures:
6 of {4,6}*48a
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143);;
s1 := ( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 17)( 6, 20)( 7, 19)( 8, 18)( 9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73,121)( 74,124)( 75,123)( 76,122)( 77,125)( 78,128)( 79,127)( 80,126)( 81,129)( 82,132)( 83,131)( 84,130)( 85,109)( 86,112)( 87,111)( 88,110)( 89,113)( 90,116)( 91,115)( 92,114)( 93,117)( 94,120)( 95,119)( 96,118)( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)(105,141)(106,144)(107,143)(108,142);;
s2 := ( 1, 74)( 2, 73)( 3, 76)( 4, 75)( 5, 82)( 6, 81)( 7, 84)( 8, 83)( 9, 78)( 10, 77)( 11, 80)( 12, 79)( 13, 86)( 14, 85)( 15, 88)( 16, 87)( 17, 94)( 18, 93)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 92)( 24, 91)( 25, 98)( 26, 97)( 27,100)( 28, 99)( 29,106)( 30,105)( 31,108)( 32,107)( 33,102)( 34,101)( 35,104)( 36,103)( 37,110)( 38,109)( 39,112)( 40,111)( 41,118)( 42,117)( 43,120)( 44,119)( 45,114)( 46,113)( 47,116)( 48,115)( 49,122)( 50,121)( 51,124)( 52,123)( 53,130)( 54,129)( 55,132)( 56,131)( 57,126)( 58,125)( 59,128)( 60,127)( 61,134)( 62,133)( 63,136)( 64,135)( 65,142)( 66,141)( 67,144)( 68,143)( 69,138)( 70,137)( 71,140)( 72,139);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 97,101)( 98,102)( 99,103)(100,104)(109,113)(110,114)(111,115)(112,116)(121,125)(122,126)(123,127)(124,128)(133,137)(134,138)(135,139)(136,140);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(144)!( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143);
s1 := Sym(144)!( 1, 13)( 2, 16)( 3, 15)( 4, 14)( 5, 17)( 6, 20)( 7, 19)( 8, 18)( 9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73,121)( 74,124)( 75,123)( 76,122)( 77,125)( 78,128)( 79,127)( 80,126)( 81,129)( 82,132)( 83,131)( 84,130)( 85,109)( 86,112)( 87,111)( 88,110)( 89,113)( 90,116)( 91,115)( 92,114)( 93,117)( 94,120)( 95,119)( 96,118)( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)(105,141)(106,144)(107,143)(108,142);
s2 := Sym(144)!( 1, 74)( 2, 73)( 3, 76)( 4, 75)( 5, 82)( 6, 81)( 7, 84)( 8, 83)( 9, 78)( 10, 77)( 11, 80)( 12, 79)( 13, 86)( 14, 85)( 15, 88)( 16, 87)( 17, 94)( 18, 93)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 92)( 24, 91)( 25, 98)( 26, 97)( 27,100)( 28, 99)( 29,106)( 30,105)( 31,108)( 32,107)( 33,102)( 34,101)( 35,104)( 36,103)( 37,110)( 38,109)( 39,112)( 40,111)( 41,118)( 42,117)( 43,120)( 44,119)( 45,114)( 46,113)( 47,116)( 48,115)( 49,122)( 50,121)( 51,124)( 52,123)( 53,130)( 54,129)( 55,132)( 56,131)( 57,126)( 58,125)( 59,128)( 60,127)( 61,134)( 62,133)( 63,136)( 64,135)( 65,142)( 66,141)( 67,144)( 68,143)( 69,138)( 70,137)( 71,140)( 72,139);
s3 := Sym(144)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 97,101)( 98,102)( 99,103)(100,104)(109,113)(110,114)(111,115)(112,116)(121,125)(122,126)(123,127)(124,128)(133,137)(134,138)(135,139)(136,140);
poly := sub<Sym(144)|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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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