Polytope of Type {2,6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,4}*768
if this polytope has a name.
Group : SmallGroup(768,1090146)
Rank : 5
Schlafli Type : {2,6,6,4}
Number of vertices, edges, etc : 2, 8, 24, 16, 4
Order of s0s1s2s3s4 : 4
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 : {2,3,6,4}*384, {2,6,6,2}*384
4-fold quotients : {2,3,6,2}*192, {2,6,3,2}*192
8-fold quotients : {2,3,3,2}*96
12-fold quotients : {2,2,2,4}*64
24-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 40, 41)
( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 52, 53)( 55, 59)( 56, 61)( 57, 60)
( 58, 62)( 64, 65)( 67, 71)( 68, 73)( 69, 72)( 70, 74)( 76, 77)( 79, 83)
( 80, 85)( 81, 84)( 82, 86)( 88, 89)( 91, 95)( 92, 97)( 93, 96)( 94, 98)
(100,101)(103,107)(104,109)(105,108)(106,110)(112,113)(115,119)(116,121)
(117,120)(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)
(139,143)(140,145)(141,144)(142,146)(148,149)(151,155)(152,157)(153,156)
(154,158)(160,161)(163,167)(164,169)(165,168)(166,170)(172,173)(175,179)
(176,181)(177,180)(178,182)(184,185)(187,191)(188,193)(189,192)(190,194);;
s2 := ( 3,103)( 4,104)( 5,106)( 6,105)( 7, 99)( 8,100)( 9,102)( 10,101)
( 11,107)( 12,108)( 13,110)( 14,109)( 15,115)( 16,116)( 17,118)( 18,117)
( 19,111)( 20,112)( 21,114)( 22,113)( 23,119)( 24,120)( 25,122)( 26,121)
( 27,127)( 28,128)( 29,130)( 30,129)( 31,123)( 32,124)( 33,126)( 34,125)
( 35,131)( 36,132)( 37,134)( 38,133)( 39,139)( 40,140)( 41,142)( 42,141)
( 43,135)( 44,136)( 45,138)( 46,137)( 47,143)( 48,144)( 49,146)( 50,145)
( 51,151)( 52,152)( 53,154)( 54,153)( 55,147)( 56,148)( 57,150)( 58,149)
( 59,155)( 60,156)( 61,158)( 62,157)( 63,163)( 64,164)( 65,166)( 66,165)
( 67,159)( 68,160)( 69,162)( 70,161)( 71,167)( 72,168)( 73,170)( 74,169)
( 75,175)( 76,176)( 77,178)( 78,177)( 79,171)( 80,172)( 81,174)( 82,173)
( 83,179)( 84,180)( 85,182)( 86,181)( 87,187)( 88,188)( 89,190)( 90,189)
( 91,183)( 92,184)( 93,186)( 94,185)( 95,191)( 96,192)( 97,194)( 98,193);;
s3 := ( 3, 54)( 4, 52)( 5, 53)( 6, 51)( 7, 62)( 8, 60)( 9, 61)( 10, 59)
( 11, 58)( 12, 56)( 13, 57)( 14, 55)( 15, 66)( 16, 64)( 17, 65)( 18, 63)
( 19, 74)( 20, 72)( 21, 73)( 22, 71)( 23, 70)( 24, 68)( 25, 69)( 26, 67)
( 27, 90)( 28, 88)( 29, 89)( 30, 87)( 31, 98)( 32, 96)( 33, 97)( 34, 95)
( 35, 94)( 36, 92)( 37, 93)( 38, 91)( 39, 78)( 40, 76)( 41, 77)( 42, 75)
( 43, 86)( 44, 84)( 45, 85)( 46, 83)( 47, 82)( 48, 80)( 49, 81)( 50, 79)
( 99,150)(100,148)(101,149)(102,147)(103,158)(104,156)(105,157)(106,155)
(107,154)(108,152)(109,153)(110,151)(111,162)(112,160)(113,161)(114,159)
(115,170)(116,168)(117,169)(118,167)(119,166)(120,164)(121,165)(122,163)
(123,186)(124,184)(125,185)(126,183)(127,194)(128,192)(129,193)(130,191)
(131,190)(132,188)(133,189)(134,187)(135,174)(136,172)(137,173)(138,171)
(139,182)(140,180)(141,181)(142,179)(143,178)(144,176)(145,177)(146,175);;
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, 39)( 16, 40)( 17, 41)( 18, 42)
( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)( 24, 48)( 25, 49)( 26, 50)
( 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, 87)( 64, 88)( 65, 89)( 66, 90)
( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 96)( 73, 97)( 74, 98)
( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)
(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)
(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)(121,145)(122,146)
(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)
(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)
(163,187)(164,188)(165,189)(166,190)(167,191)(168,192)(169,193)(170,194);;
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(194)!(1,2);
s1 := Sym(194)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 40, 41)( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 52, 53)( 55, 59)( 56, 61)
( 57, 60)( 58, 62)( 64, 65)( 67, 71)( 68, 73)( 69, 72)( 70, 74)( 76, 77)
( 79, 83)( 80, 85)( 81, 84)( 82, 86)( 88, 89)( 91, 95)( 92, 97)( 93, 96)
( 94, 98)(100,101)(103,107)(104,109)(105,108)(106,110)(112,113)(115,119)
(116,121)(117,120)(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)
(136,137)(139,143)(140,145)(141,144)(142,146)(148,149)(151,155)(152,157)
(153,156)(154,158)(160,161)(163,167)(164,169)(165,168)(166,170)(172,173)
(175,179)(176,181)(177,180)(178,182)(184,185)(187,191)(188,193)(189,192)
(190,194);
s2 := Sym(194)!( 3,103)( 4,104)( 5,106)( 6,105)( 7, 99)( 8,100)( 9,102)
( 10,101)( 11,107)( 12,108)( 13,110)( 14,109)( 15,115)( 16,116)( 17,118)
( 18,117)( 19,111)( 20,112)( 21,114)( 22,113)( 23,119)( 24,120)( 25,122)
( 26,121)( 27,127)( 28,128)( 29,130)( 30,129)( 31,123)( 32,124)( 33,126)
( 34,125)( 35,131)( 36,132)( 37,134)( 38,133)( 39,139)( 40,140)( 41,142)
( 42,141)( 43,135)( 44,136)( 45,138)( 46,137)( 47,143)( 48,144)( 49,146)
( 50,145)( 51,151)( 52,152)( 53,154)( 54,153)( 55,147)( 56,148)( 57,150)
( 58,149)( 59,155)( 60,156)( 61,158)( 62,157)( 63,163)( 64,164)( 65,166)
( 66,165)( 67,159)( 68,160)( 69,162)( 70,161)( 71,167)( 72,168)( 73,170)
( 74,169)( 75,175)( 76,176)( 77,178)( 78,177)( 79,171)( 80,172)( 81,174)
( 82,173)( 83,179)( 84,180)( 85,182)( 86,181)( 87,187)( 88,188)( 89,190)
( 90,189)( 91,183)( 92,184)( 93,186)( 94,185)( 95,191)( 96,192)( 97,194)
( 98,193);
s3 := Sym(194)!( 3, 54)( 4, 52)( 5, 53)( 6, 51)( 7, 62)( 8, 60)( 9, 61)
( 10, 59)( 11, 58)( 12, 56)( 13, 57)( 14, 55)( 15, 66)( 16, 64)( 17, 65)
( 18, 63)( 19, 74)( 20, 72)( 21, 73)( 22, 71)( 23, 70)( 24, 68)( 25, 69)
( 26, 67)( 27, 90)( 28, 88)( 29, 89)( 30, 87)( 31, 98)( 32, 96)( 33, 97)
( 34, 95)( 35, 94)( 36, 92)( 37, 93)( 38, 91)( 39, 78)( 40, 76)( 41, 77)
( 42, 75)( 43, 86)( 44, 84)( 45, 85)( 46, 83)( 47, 82)( 48, 80)( 49, 81)
( 50, 79)( 99,150)(100,148)(101,149)(102,147)(103,158)(104,156)(105,157)
(106,155)(107,154)(108,152)(109,153)(110,151)(111,162)(112,160)(113,161)
(114,159)(115,170)(116,168)(117,169)(118,167)(119,166)(120,164)(121,165)
(122,163)(123,186)(124,184)(125,185)(126,183)(127,194)(128,192)(129,193)
(130,191)(131,190)(132,188)(133,189)(134,187)(135,174)(136,172)(137,173)
(138,171)(139,182)(140,180)(141,181)(142,179)(143,178)(144,176)(145,177)
(146,175);
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, 39)( 16, 40)( 17, 41)
( 18, 42)( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)( 24, 48)( 25, 49)
( 26, 50)( 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, 87)( 64, 88)( 65, 89)
( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 96)( 73, 97)
( 74, 98)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)(105,129)
(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)
(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)(121,145)
(122,146)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)
(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)(161,185)
(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192)(169,193)
(170,194);
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 >;
to this polytope