Polytope of Type {6,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,4}*768b
if this polytope has a name.
Group : SmallGroup(768,1087581)
Rank : 4
Schlafli Type : {6,4,4}
Number of vertices, edges, etc : 12, 48, 32, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
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 : {3,4,4}*384b
4-fold quotients : {3,4,4}*192a
8-fold quotients : {3,4,2}*96
16-fold quotients : {3,4,2}*48
32-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)
(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)
(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)
(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s1 := ( 1, 81)( 2, 83)( 3, 82)( 4, 84)( 5, 89)( 6, 91)( 7, 90)( 8, 92)
( 9, 85)( 10, 87)( 11, 86)( 12, 88)( 13, 93)( 14, 95)( 15, 94)( 16, 96)
( 17, 65)( 18, 67)( 19, 66)( 20, 68)( 21, 73)( 22, 75)( 23, 74)( 24, 76)
( 25, 69)( 26, 71)( 27, 70)( 28, 72)( 29, 77)( 30, 79)( 31, 78)( 32, 80)
( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 57)( 38, 59)( 39, 58)( 40, 60)
( 41, 53)( 42, 55)( 43, 54)( 44, 56)( 45, 61)( 46, 63)( 47, 62)( 48, 64)
( 97,177)( 98,179)( 99,178)(100,180)(101,185)(102,187)(103,186)(104,188)
(105,181)(106,183)(107,182)(108,184)(109,189)(110,191)(111,190)(112,192)
(113,161)(114,163)(115,162)(116,164)(117,169)(118,171)(119,170)(120,172)
(121,165)(122,167)(123,166)(124,168)(125,173)(126,175)(127,174)(128,176)
(129,145)(130,147)(131,146)(132,148)(133,153)(134,155)(135,154)(136,156)
(137,149)(138,151)(139,150)(140,152)(141,157)(142,159)(143,158)(144,160);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)(104,148)
(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)
(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)
(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)
(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)
(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)( 8,103)
( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)( 16,109)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)( 24,119)
( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)( 32,125)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)( 40,135)
( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)( 56,151)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)( 64,157)
( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)( 72,167)
( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)( 80,173)
( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)( 88,183)
( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)( 96,189);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s3*s2*s1*s2*s0*s1*s2*s3*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)
(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)
(118,134)(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)
(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)
(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)
(176,187);
s1 := Sym(192)!( 1, 81)( 2, 83)( 3, 82)( 4, 84)( 5, 89)( 6, 91)( 7, 90)
( 8, 92)( 9, 85)( 10, 87)( 11, 86)( 12, 88)( 13, 93)( 14, 95)( 15, 94)
( 16, 96)( 17, 65)( 18, 67)( 19, 66)( 20, 68)( 21, 73)( 22, 75)( 23, 74)
( 24, 76)( 25, 69)( 26, 71)( 27, 70)( 28, 72)( 29, 77)( 30, 79)( 31, 78)
( 32, 80)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 41, 53)( 42, 55)( 43, 54)( 44, 56)( 45, 61)( 46, 63)( 47, 62)
( 48, 64)( 97,177)( 98,179)( 99,178)(100,180)(101,185)(102,187)(103,186)
(104,188)(105,181)(106,183)(107,182)(108,184)(109,189)(110,191)(111,190)
(112,192)(113,161)(114,163)(115,162)(116,164)(117,169)(118,171)(119,170)
(120,172)(121,165)(122,167)(123,166)(124,168)(125,173)(126,175)(127,174)
(128,176)(129,145)(130,147)(131,146)(132,148)(133,153)(134,155)(135,154)
(136,156)(137,149)(138,151)(139,150)(140,152)(141,157)(142,159)(143,158)
(144,160);
s2 := Sym(192)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)
(104,148)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)
(112,156)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)
(120,164)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)
(128,172)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)
(136,180)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)
(144,188);
s3 := Sym(192)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)
( 8,103)( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)
( 16,109)( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)
( 24,119)( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)
( 32,125)( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)
( 40,135)( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)
( 56,151)( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)
( 64,157)( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)
( 72,167)( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)
( 80,173)( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)
( 88,183)( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)
( 96,189);
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,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s3*s2*s1*s2*s0*s1*s2*s3*s2*s1*s2*s1*s0*s1 >;
References : None.
to this polytope