Polytope of Type {2,4,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6,8}*768a
if this polytope has a name.
Group : SmallGroup(768,1044756)
Rank : 5
Schlafli Type : {2,4,6,8}
Number of vertices, edges, etc : 2, 4, 12, 24, 8
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,4,6,4}*384a, {2,2,6,8}*384
3-fold quotients : {2,4,2,8}*256
4-fold quotients : {2,2,6,4}*192a, {2,4,6,2}*192a
6-fold quotients : {2,4,2,4}*128, {2,2,2,8}*128
8-fold quotients : {2,2,6,2}*96
12-fold quotients : {2,2,2,4}*64, {2,4,2,2}*64
16-fold quotients : {2,2,3,2}*48
24-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)( 9, 57)( 10, 58)
( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)( 17, 65)( 18, 66)
( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)( 25, 73)( 26, 74)
( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)( 33, 81)( 34, 82)
( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)( 40, 88)( 41, 89)( 42, 90)
( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)( 49, 97)( 50, 98)
( 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)(145,193)(146,194);;
s2 := ( 3, 39)( 4, 41)( 5, 40)( 6, 42)( 7, 44)( 8, 43)( 9, 45)( 10, 47)
( 11, 46)( 12, 48)( 13, 50)( 14, 49)( 15, 30)( 16, 32)( 17, 31)( 18, 27)
( 19, 29)( 20, 28)( 21, 36)( 22, 38)( 23, 37)( 24, 33)( 25, 35)( 26, 34)
( 51, 93)( 52, 95)( 53, 94)( 54, 96)( 55, 98)( 56, 97)( 57, 87)( 58, 89)
( 59, 88)( 60, 90)( 61, 92)( 62, 91)( 63, 84)( 64, 86)( 65, 85)( 66, 81)
( 67, 83)( 68, 82)( 69, 78)( 70, 80)( 71, 79)( 72, 75)( 73, 77)( 74, 76)
( 99,123)(100,125)(101,124)(102,126)(103,128)(104,127)(105,129)(106,131)
(107,130)(108,132)(109,134)(110,133)(111,135)(112,137)(113,136)(114,138)
(115,140)(116,139)(117,141)(118,143)(119,142)(120,144)(121,146)(122,145)
(147,177)(148,179)(149,178)(150,180)(151,182)(152,181)(153,171)(154,173)
(155,172)(156,174)(157,176)(158,175)(159,189)(160,191)(161,190)(162,192)
(163,194)(164,193)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187);;
s3 := ( 3,100)( 4, 99)( 5,101)( 6,103)( 7,102)( 8,104)( 9,106)( 10,105)
( 11,107)( 12,109)( 13,108)( 14,110)( 15,115)( 16,114)( 17,116)( 18,112)
( 19,111)( 20,113)( 21,121)( 22,120)( 23,122)( 24,118)( 25,117)( 26,119)
( 27,136)( 28,135)( 29,137)( 30,139)( 31,138)( 32,140)( 33,142)( 34,141)
( 35,143)( 36,145)( 37,144)( 38,146)( 39,124)( 40,123)( 41,125)( 42,127)
( 43,126)( 44,128)( 45,130)( 46,129)( 47,131)( 48,133)( 49,132)( 50,134)
( 51,148)( 52,147)( 53,149)( 54,151)( 55,150)( 56,152)( 57,154)( 58,153)
( 59,155)( 60,157)( 61,156)( 62,158)( 63,163)( 64,162)( 65,164)( 66,160)
( 67,159)( 68,161)( 69,169)( 70,168)( 71,170)( 72,166)( 73,165)( 74,167)
( 75,184)( 76,183)( 77,185)( 78,187)( 79,186)( 80,188)( 81,190)( 82,189)
( 83,191)( 84,193)( 85,192)( 86,194)( 87,172)( 88,171)( 89,173)( 90,175)
( 91,174)( 92,176)( 93,178)( 94,177)( 95,179)( 96,181)( 97,180)( 98,182);;
s4 := ( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)( 9, 33)( 10, 34)
( 11, 35)( 12, 36)( 13, 37)( 14, 38)( 15, 42)( 16, 43)( 17, 44)( 18, 39)
( 19, 40)( 20, 41)( 21, 48)( 22, 49)( 23, 50)( 24, 45)( 25, 46)( 26, 47)
( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 81)( 58, 82)
( 59, 83)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)( 65, 92)( 66, 87)
( 67, 88)( 68, 89)( 69, 96)( 70, 97)( 71, 98)( 72, 93)( 73, 94)( 74, 95)
( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)
(107,131)(108,132)(109,133)(110,134)(111,138)(112,139)(113,140)(114,135)
(115,136)(116,137)(117,144)(118,145)(119,146)(120,141)(121,142)(122,143)
(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)
(155,179)(156,180)(157,181)(158,182)(159,186)(160,187)(161,188)(162,183)
(163,184)(164,185)(165,192)(166,193)(167,194)(168,189)(169,190)(170,191);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(194)!(1,2);
s1 := Sym(194)!( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)( 9, 57)
( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)( 17, 65)
( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)( 25, 73)
( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)( 33, 81)
( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)( 40, 88)( 41, 89)
( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)( 49, 97)
( 50, 98)( 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)(145,193)
(146,194);
s2 := Sym(194)!( 3, 39)( 4, 41)( 5, 40)( 6, 42)( 7, 44)( 8, 43)( 9, 45)
( 10, 47)( 11, 46)( 12, 48)( 13, 50)( 14, 49)( 15, 30)( 16, 32)( 17, 31)
( 18, 27)( 19, 29)( 20, 28)( 21, 36)( 22, 38)( 23, 37)( 24, 33)( 25, 35)
( 26, 34)( 51, 93)( 52, 95)( 53, 94)( 54, 96)( 55, 98)( 56, 97)( 57, 87)
( 58, 89)( 59, 88)( 60, 90)( 61, 92)( 62, 91)( 63, 84)( 64, 86)( 65, 85)
( 66, 81)( 67, 83)( 68, 82)( 69, 78)( 70, 80)( 71, 79)( 72, 75)( 73, 77)
( 74, 76)( 99,123)(100,125)(101,124)(102,126)(103,128)(104,127)(105,129)
(106,131)(107,130)(108,132)(109,134)(110,133)(111,135)(112,137)(113,136)
(114,138)(115,140)(116,139)(117,141)(118,143)(119,142)(120,144)(121,146)
(122,145)(147,177)(148,179)(149,178)(150,180)(151,182)(152,181)(153,171)
(154,173)(155,172)(156,174)(157,176)(158,175)(159,189)(160,191)(161,190)
(162,192)(163,194)(164,193)(165,183)(166,185)(167,184)(168,186)(169,188)
(170,187);
s3 := Sym(194)!( 3,100)( 4, 99)( 5,101)( 6,103)( 7,102)( 8,104)( 9,106)
( 10,105)( 11,107)( 12,109)( 13,108)( 14,110)( 15,115)( 16,114)( 17,116)
( 18,112)( 19,111)( 20,113)( 21,121)( 22,120)( 23,122)( 24,118)( 25,117)
( 26,119)( 27,136)( 28,135)( 29,137)( 30,139)( 31,138)( 32,140)( 33,142)
( 34,141)( 35,143)( 36,145)( 37,144)( 38,146)( 39,124)( 40,123)( 41,125)
( 42,127)( 43,126)( 44,128)( 45,130)( 46,129)( 47,131)( 48,133)( 49,132)
( 50,134)( 51,148)( 52,147)( 53,149)( 54,151)( 55,150)( 56,152)( 57,154)
( 58,153)( 59,155)( 60,157)( 61,156)( 62,158)( 63,163)( 64,162)( 65,164)
( 66,160)( 67,159)( 68,161)( 69,169)( 70,168)( 71,170)( 72,166)( 73,165)
( 74,167)( 75,184)( 76,183)( 77,185)( 78,187)( 79,186)( 80,188)( 81,190)
( 82,189)( 83,191)( 84,193)( 85,192)( 86,194)( 87,172)( 88,171)( 89,173)
( 90,175)( 91,174)( 92,176)( 93,178)( 94,177)( 95,179)( 96,181)( 97,180)
( 98,182);
s4 := Sym(194)!( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)( 9, 33)
( 10, 34)( 11, 35)( 12, 36)( 13, 37)( 14, 38)( 15, 42)( 16, 43)( 17, 44)
( 18, 39)( 19, 40)( 20, 41)( 21, 48)( 22, 49)( 23, 50)( 24, 45)( 25, 46)
( 26, 47)( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)( 57, 81)
( 58, 82)( 59, 83)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)( 65, 92)
( 66, 87)( 67, 88)( 68, 89)( 69, 96)( 70, 97)( 71, 98)( 72, 93)( 73, 94)
( 74, 95)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)
(106,130)(107,131)(108,132)(109,133)(110,134)(111,138)(112,139)(113,140)
(114,135)(115,136)(116,137)(117,144)(118,145)(119,146)(120,141)(121,142)
(122,143)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)
(154,178)(155,179)(156,180)(157,181)(158,182)(159,186)(160,187)(161,188)
(162,183)(163,184)(164,185)(165,192)(166,193)(167,194)(168,189)(169,190)
(170,191);
poly := sub<Sym(194)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s4*s3*s2*s3*s4*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope