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