Polytope of Type {4,6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,8}*768g
if this polytope has a name.
Group : SmallGroup(768,1090200)
Rank : 4
Schlafli Type : {4,6,8}
Number of vertices, edges, etc : 4, 24, 48, 16
Order of s₀s₁s₂s₃ : 6
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 : {4,6,4}*384c
   4-fold quotients : {4,3,4}*192a, {4,6,4}*192d, {4,6,4}*192e
   8-fold quotients : {4,6,2}*96c, {4,3,4}*96
   16-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope