Polytope of Type {4,8,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,6}*384a
Also Known As : {{4,8|2},{8,6|2}}. if this polytope has another name.
Group : SmallGroup(384,12576)
Rank : 4
Schlafli Type : {4,8,6}
Number of vertices, edges, etc : 4, 16, 24, 6
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,8,6,2} of size 768
{4,8,6,3} of size 1152
Vertex Figure Of :
{2,4,8,6} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,6}*192, {2,8,6}*192
3-fold quotients : {4,8,2}*128a
4-fold quotients : {2,4,6}*96a, {4,2,6}*96
6-fold quotients : {4,4,2}*64, {2,8,2}*64
8-fold quotients : {4,2,3}*48, {2,2,6}*48
12-fold quotients : {2,4,2}*32, {4,2,2}*32
16-fold quotients : {2,2,3}*24
24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8,6}*768a, {8,8,6}*768a, {8,8,6}*768b, {4,8,12}*768d, {4,16,6}*768a, {4,16,6}*768b
3-fold covers : {4,8,18}*1152a, {12,8,6}*1152a, {4,24,6}*1152a, {4,24,6}*1152c
5-fold covers : {4,8,30}*1920a, {20,8,6}*1920a, {4,40,6}*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 := ( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 37, 40)( 38, 41)
( 39, 42)( 43, 46)( 44, 47)( 45, 48)( 49, 55)( 50, 56)( 51, 57)( 52, 58)
( 53, 59)( 54, 60)( 61, 70)( 62, 71)( 63, 72)( 64, 67)( 65, 68)( 66, 69)
( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 94)( 86, 95)
( 87, 96)( 88, 91)( 89, 92)( 90, 93)( 97,109)( 98,110)( 99,111)(100,112)
(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120)
(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)
(129,141)(130,142)(131,143)(132,144)(145,163)(146,164)(147,165)(148,166)
(149,167)(150,168)(151,157)(152,158)(153,159)(154,160)(155,161)(156,162)
(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)(176,182)
(177,183)(178,184)(179,185)(180,186);;
s2 := ( 1,121)( 2,123)( 3,122)( 4,124)( 5,126)( 6,125)( 7,127)( 8,129)
( 9,128)( 10,130)( 11,132)( 12,131)( 13,136)( 14,138)( 15,137)( 16,133)
( 17,135)( 18,134)( 19,142)( 20,144)( 21,143)( 22,139)( 23,141)( 24,140)
( 25, 97)( 26, 99)( 27, 98)( 28,100)( 29,102)( 30,101)( 31,103)( 32,105)
( 33,104)( 34,106)( 35,108)( 36,107)( 37,112)( 38,114)( 39,113)( 40,109)
( 41,111)( 42,110)( 43,118)( 44,120)( 45,119)( 46,115)( 47,117)( 48,116)
( 49,169)( 50,171)( 51,170)( 52,172)( 53,174)( 54,173)( 55,175)( 56,177)
( 57,176)( 58,178)( 59,180)( 60,179)( 61,184)( 62,186)( 63,185)( 64,181)
( 65,183)( 66,182)( 67,190)( 68,192)( 69,191)( 70,187)( 71,189)( 72,188)
( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,151)( 80,153)
( 81,152)( 82,154)( 83,156)( 84,155)( 85,160)( 86,162)( 87,161)( 88,157)
( 89,159)( 90,158)( 91,166)( 92,168)( 93,167)( 94,163)( 95,165)( 96,164);;
s3 := ( 1, 26)( 2, 25)( 3, 27)( 4, 29)( 5, 28)( 6, 30)( 7, 32)( 8, 31)
( 9, 33)( 10, 35)( 11, 34)( 12, 36)( 13, 38)( 14, 37)( 15, 39)( 16, 41)
( 17, 40)( 18, 42)( 19, 44)( 20, 43)( 21, 45)( 22, 47)( 23, 46)( 24, 48)
( 49, 74)( 50, 73)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 80)( 56, 79)
( 57, 81)( 58, 83)( 59, 82)( 60, 84)( 61, 86)( 62, 85)( 63, 87)( 64, 89)
( 65, 88)( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 95)( 71, 94)( 72, 96)
( 97,122)( 98,121)( 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)
(145,170)(146,169)(147,171)(148,173)(149,172)(150,174)(151,176)(152,175)
(153,177)(154,179)(155,178)(156,180)(157,182)(158,181)(159,183)(160,185)
(161,184)(162,186)(163,188)(164,187)(165,189)(166,191)(167,190)(168,192);;
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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)!( 13, 16)( 14, 17)( 15, 18)( 19, 22)( 20, 23)( 21, 24)( 37, 40)
( 38, 41)( 39, 42)( 43, 46)( 44, 47)( 45, 48)( 49, 55)( 50, 56)( 51, 57)
( 52, 58)( 53, 59)( 54, 60)( 61, 70)( 62, 71)( 63, 72)( 64, 67)( 65, 68)
( 66, 69)( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 94)
( 86, 95)( 87, 96)( 88, 91)( 89, 92)( 90, 93)( 97,109)( 98,110)( 99,111)
(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)
(108,120)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)
(128,140)(129,141)(130,142)(131,143)(132,144)(145,163)(146,164)(147,165)
(148,166)(149,167)(150,168)(151,157)(152,158)(153,159)(154,160)(155,161)
(156,162)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)
(176,182)(177,183)(178,184)(179,185)(180,186);
s2 := Sym(192)!( 1,121)( 2,123)( 3,122)( 4,124)( 5,126)( 6,125)( 7,127)
( 8,129)( 9,128)( 10,130)( 11,132)( 12,131)( 13,136)( 14,138)( 15,137)
( 16,133)( 17,135)( 18,134)( 19,142)( 20,144)( 21,143)( 22,139)( 23,141)
( 24,140)( 25, 97)( 26, 99)( 27, 98)( 28,100)( 29,102)( 30,101)( 31,103)
( 32,105)( 33,104)( 34,106)( 35,108)( 36,107)( 37,112)( 38,114)( 39,113)
( 40,109)( 41,111)( 42,110)( 43,118)( 44,120)( 45,119)( 46,115)( 47,117)
( 48,116)( 49,169)( 50,171)( 51,170)( 52,172)( 53,174)( 54,173)( 55,175)
( 56,177)( 57,176)( 58,178)( 59,180)( 60,179)( 61,184)( 62,186)( 63,185)
( 64,181)( 65,183)( 66,182)( 67,190)( 68,192)( 69,191)( 70,187)( 71,189)
( 72,188)( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,151)
( 80,153)( 81,152)( 82,154)( 83,156)( 84,155)( 85,160)( 86,162)( 87,161)
( 88,157)( 89,159)( 90,158)( 91,166)( 92,168)( 93,167)( 94,163)( 95,165)
( 96,164);
s3 := Sym(192)!( 1, 26)( 2, 25)( 3, 27)( 4, 29)( 5, 28)( 6, 30)( 7, 32)
( 8, 31)( 9, 33)( 10, 35)( 11, 34)( 12, 36)( 13, 38)( 14, 37)( 15, 39)
( 16, 41)( 17, 40)( 18, 42)( 19, 44)( 20, 43)( 21, 45)( 22, 47)( 23, 46)
( 24, 48)( 49, 74)( 50, 73)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 80)
( 56, 79)( 57, 81)( 58, 83)( 59, 82)( 60, 84)( 61, 86)( 62, 85)( 63, 87)
( 64, 89)( 65, 88)( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 95)( 71, 94)
( 72, 96)( 97,122)( 98,121)( 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)(145,170)(146,169)(147,171)(148,173)(149,172)(150,174)(151,176)
(152,175)(153,177)(154,179)(155,178)(156,180)(157,182)(158,181)(159,183)
(160,185)(161,184)(162,186)(163,188)(164,187)(165,189)(166,191)(167,190)
(168,192);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope