# Polytope of Type {4,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,3}*768b
Also Known As : {{4,6|2},{6,3}8}. if this polytope has another name.
Group : SmallGroup(768,1087795)
Rank : 4
Schlafli Type : {4,6,3}
Number of vertices, edges, etc : 4, 64, 48, 16
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {2,6,3}*384
4-fold quotients : {4,6,3}*192
8-fold quotients : {2,6,3}*96
16-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 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, 97)(  2, 98)(  3,100)(  4, 99)(  5,101)(  6,102)(  7,104)(  8,103)
(  9,112)( 10,111)( 11,109)( 12,110)( 13,107)( 14,108)( 15,106)( 16,105)
( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)( 24,135)
( 25,144)( 26,143)( 27,141)( 28,142)( 29,139)( 30,140)( 31,138)( 32,137)
( 33,113)( 34,114)( 35,116)( 36,115)( 37,117)( 38,118)( 39,120)( 40,119)
( 41,128)( 42,127)( 43,125)( 44,126)( 45,123)( 46,124)( 47,122)( 48,121)
( 49,145)( 50,146)( 51,148)( 52,147)( 53,149)( 54,150)( 55,152)( 56,151)
( 57,160)( 58,159)( 59,157)( 60,158)( 61,155)( 62,156)( 63,154)( 64,153)
( 65,177)( 66,178)( 67,180)( 68,179)( 69,181)( 70,182)( 71,184)( 72,183)
( 73,192)( 74,191)( 75,189)( 76,190)( 77,187)( 78,188)( 79,186)( 80,185)
( 81,161)( 82,162)( 83,164)( 84,163)( 85,165)( 86,166)( 87,168)( 88,167)
( 89,176)( 90,175)( 91,173)( 92,174)( 93,171)( 94,172)( 95,170)( 96,169);;
s2 := (  1, 33)(  2, 35)(  3, 34)(  4, 36)(  5, 44)(  6, 42)(  7, 43)(  8, 41)
(  9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)( 16, 48)
( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)( 50, 83)
( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)( 58, 86)
( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 76)
( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)
(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)(108,133)
(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)(119,123)
(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)(150,186)
(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)(158,191)
(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)(174,175);;
s3 := (  1,  6)(  2,  5)(  3,  7)(  4,  8)( 11, 12)( 13, 14)( 17, 38)( 18, 37)
( 19, 39)( 20, 40)( 21, 34)( 22, 33)( 23, 35)( 24, 36)( 25, 41)( 26, 42)
( 27, 44)( 28, 43)( 29, 46)( 30, 45)( 31, 47)( 32, 48)( 49, 54)( 50, 53)
( 51, 55)( 52, 56)( 59, 60)( 61, 62)( 65, 86)( 66, 85)( 67, 87)( 68, 88)
( 69, 82)( 70, 81)( 71, 83)( 72, 84)( 73, 89)( 74, 90)( 75, 92)( 76, 91)
( 77, 94)( 78, 93)( 79, 95)( 80, 96)( 97,102)( 98,101)( 99,103)(100,104)
(107,108)(109,110)(113,134)(114,133)(115,135)(116,136)(117,130)(118,129)
(119,131)(120,132)(121,137)(122,138)(123,140)(124,139)(125,142)(126,141)
(127,143)(128,144)(145,150)(146,149)(147,151)(148,152)(155,156)(157,158)
(161,182)(162,181)(163,183)(164,184)(165,178)(166,177)(167,179)(168,180)
(169,185)(170,186)(171,188)(172,187)(173,190)(174,189)(175,191)(176,192);;
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, s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope