Polytope of Type {4,6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,8}*384a
Also Known As : {{4,6|2},{6,8|2}}. if this polytope has another name.
Group : SmallGroup(384,12600)
Rank : 4
Schlafli Type : {4,6,8}
Number of vertices, edges, etc : 4, 12, 24, 8
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,8,2} of size 768
Vertex Figure Of :
   {2,4,6,8} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,4}*192a, {2,6,8}*192
   3-fold quotients : {4,2,8}*128
   4-fold quotients : {2,6,4}*96a, {4,6,2}*96a
   6-fold quotients : {4,2,4}*64, {2,2,8}*64
   8-fold quotients : {2,6,2}*48
   12-fold quotients : {2,2,4}*32, {4,2,2}*32
   16-fold quotients : {2,3,2}*24
   24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,6,8}*768, {4,12,8}*768a, {4,6,16}*768a
   3-fold covers : {4,18,8}*1152a, {12,6,8}*1152a, {12,6,8}*1152b, {4,6,24}*1152a, {4,6,24}*1152b
   5-fold covers : {4,30,8}*1920a, {20,6,8}*1920, {4,6,40}*1920a
Permutation Representation (GAP) :

s0 := (  1, 49)(  2, 50)(  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)
( 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);;
s1 := (  1, 37)(  2, 39)(  3, 38)(  4, 40)(  5, 42)(  6, 41)(  7, 43)(  8, 45)
(  9, 44)( 10, 46)( 11, 48)( 12, 47)( 13, 28)( 14, 30)( 15, 29)( 16, 25)
( 17, 27)( 18, 26)( 19, 34)( 20, 36)( 21, 35)( 22, 31)( 23, 33)( 24, 32)
( 49, 91)( 50, 93)( 51, 92)( 52, 94)( 53, 96)( 54, 95)( 55, 85)( 56, 87)
( 57, 86)( 58, 88)( 59, 90)( 60, 89)( 61, 82)( 62, 84)( 63, 83)( 64, 79)
( 65, 81)( 66, 80)( 67, 76)( 68, 78)( 69, 77)( 70, 73)( 71, 75)( 72, 74)
( 97,121)( 98,123)( 99,122)(100,124)(101,126)(102,125)(103,127)(104,129)
(105,128)(106,130)(107,132)(108,131)(109,133)(110,135)(111,134)(112,136)
(113,138)(114,137)(115,139)(116,141)(117,140)(118,142)(119,144)(120,143)
(145,175)(146,177)(147,176)(148,178)(149,180)(150,179)(151,169)(152,171)
(153,170)(154,172)(155,174)(156,173)(157,187)(158,189)(159,188)(160,190)
(161,192)(162,191)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185);;
s2 := (  1, 98)(  2, 97)(  3, 99)(  4,101)(  5,100)(  6,102)(  7,104)(  8,103)
(  9,105)( 10,107)( 11,106)( 12,108)( 13,113)( 14,112)( 15,114)( 16,110)
( 17,109)( 18,111)( 19,119)( 20,118)( 21,120)( 22,116)( 23,115)( 24,117)
( 25,134)( 26,133)( 27,135)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)
( 33,141)( 34,143)( 35,142)( 36,144)( 37,122)( 38,121)( 39,123)( 40,125)
( 41,124)( 42,126)( 43,128)( 44,127)( 45,129)( 46,131)( 47,130)( 48,132)
( 49,146)( 50,145)( 51,147)( 52,149)( 53,148)( 54,150)( 55,152)( 56,151)
( 57,153)( 58,155)( 59,154)( 60,156)( 61,161)( 62,160)( 63,162)( 64,158)
( 65,157)( 66,159)( 67,167)( 68,166)( 69,168)( 70,164)( 71,163)( 72,165)
( 73,182)( 74,181)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)( 80,187)
( 81,189)( 82,191)( 83,190)( 84,192)( 85,170)( 86,169)( 87,171)( 88,173)
( 89,172)( 90,174)( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)( 96,180);;
s3 := (  1, 25)(  2, 26)(  3, 27)(  4, 28)(  5, 29)(  6, 30)(  7, 31)(  8, 32)
(  9, 33)( 10, 34)( 11, 35)( 12, 36)( 13, 40)( 14, 41)( 15, 42)( 16, 37)
( 17, 38)( 18, 39)( 19, 46)( 20, 47)( 21, 48)( 22, 43)( 23, 44)( 24, 45)
( 49, 73)( 50, 74)( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)( 56, 80)
( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 88)( 62, 89)( 63, 90)( 64, 85)
( 65, 86)( 66, 87)( 67, 94)( 68, 95)( 69, 96)( 70, 91)( 71, 92)( 72, 93)
( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)
(105,129)(106,130)(107,131)(108,132)(109,136)(110,137)(111,138)(112,133)
(113,134)(114,135)(115,142)(116,143)(117,144)(118,139)(119,140)(120,141)
(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)
(153,177)(154,178)(155,179)(156,180)(157,184)(158,185)(159,186)(160,181)
(161,182)(162,183)(163,190)(164,191)(165,192)(166,187)(167,188)(168,189);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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, 49)(  2, 50)(  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)( 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);
s1 := Sym(192)!(  1, 37)(  2, 39)(  3, 38)(  4, 40)(  5, 42)(  6, 41)(  7, 43)
(  8, 45)(  9, 44)( 10, 46)( 11, 48)( 12, 47)( 13, 28)( 14, 30)( 15, 29)
( 16, 25)( 17, 27)( 18, 26)( 19, 34)( 20, 36)( 21, 35)( 22, 31)( 23, 33)
( 24, 32)( 49, 91)( 50, 93)( 51, 92)( 52, 94)( 53, 96)( 54, 95)( 55, 85)
( 56, 87)( 57, 86)( 58, 88)( 59, 90)( 60, 89)( 61, 82)( 62, 84)( 63, 83)
( 64, 79)( 65, 81)( 66, 80)( 67, 76)( 68, 78)( 69, 77)( 70, 73)( 71, 75)
( 72, 74)( 97,121)( 98,123)( 99,122)(100,124)(101,126)(102,125)(103,127)
(104,129)(105,128)(106,130)(107,132)(108,131)(109,133)(110,135)(111,134)
(112,136)(113,138)(114,137)(115,139)(116,141)(117,140)(118,142)(119,144)
(120,143)(145,175)(146,177)(147,176)(148,178)(149,180)(150,179)(151,169)
(152,171)(153,170)(154,172)(155,174)(156,173)(157,187)(158,189)(159,188)
(160,190)(161,192)(162,191)(163,181)(164,183)(165,182)(166,184)(167,186)
(168,185);
s2 := Sym(192)!(  1, 98)(  2, 97)(  3, 99)(  4,101)(  5,100)(  6,102)(  7,104)
(  8,103)(  9,105)( 10,107)( 11,106)( 12,108)( 13,113)( 14,112)( 15,114)
( 16,110)( 17,109)( 18,111)( 19,119)( 20,118)( 21,120)( 22,116)( 23,115)
( 24,117)( 25,134)( 26,133)( 27,135)( 28,137)( 29,136)( 30,138)( 31,140)
( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,122)( 38,121)( 39,123)
( 40,125)( 41,124)( 42,126)( 43,128)( 44,127)( 45,129)( 46,131)( 47,130)
( 48,132)( 49,146)( 50,145)( 51,147)( 52,149)( 53,148)( 54,150)( 55,152)
( 56,151)( 57,153)( 58,155)( 59,154)( 60,156)( 61,161)( 62,160)( 63,162)
( 64,158)( 65,157)( 66,159)( 67,167)( 68,166)( 69,168)( 70,164)( 71,163)
( 72,165)( 73,182)( 74,181)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)
( 80,187)( 81,189)( 82,191)( 83,190)( 84,192)( 85,170)( 86,169)( 87,171)
( 88,173)( 89,172)( 90,174)( 91,176)( 92,175)( 93,177)( 94,179)( 95,178)
( 96,180);
s3 := Sym(192)!(  1, 25)(  2, 26)(  3, 27)(  4, 28)(  5, 29)(  6, 30)(  7, 31)
(  8, 32)(  9, 33)( 10, 34)( 11, 35)( 12, 36)( 13, 40)( 14, 41)( 15, 42)
( 16, 37)( 17, 38)( 18, 39)( 19, 46)( 20, 47)( 21, 48)( 22, 43)( 23, 44)
( 24, 45)( 49, 73)( 50, 74)( 51, 75)( 52, 76)( 53, 77)( 54, 78)( 55, 79)
( 56, 80)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 88)( 62, 89)( 63, 90)
( 64, 85)( 65, 86)( 66, 87)( 67, 94)( 68, 95)( 69, 96)( 70, 91)( 71, 92)
( 72, 93)( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)
(104,128)(105,129)(106,130)(107,131)(108,132)(109,136)(110,137)(111,138)
(112,133)(113,134)(114,135)(115,142)(116,143)(117,144)(118,139)(119,140)
(120,141)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)
(152,176)(153,177)(154,178)(155,179)(156,180)(157,184)(158,185)(159,186)
(160,181)(161,182)(162,183)(163,190)(164,191)(165,192)(166,187)(167,188)
(168,189);
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope