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# Polytope of Type {6,4,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,8}*384a
Also Known As : {{6,4|2},{4,8|2}}. if this polytope has another name.
Group : SmallGroup(384,12576)
Rank : 4
Schlafli Type : {6,4,8}
Number of vertices, edges, etc : 6, 12, 16, 8
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,8,2} of size 768
Vertex Figure Of :
{2,6,4,8} of size 768
{3,6,4,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4,4}*192, {6,2,8}*192
3-fold quotients : {2,4,8}*128a
4-fold quotients : {3,2,8}*96, {6,2,4}*96, {6,4,2}*96a
6-fold quotients : {2,4,4}*64, {2,2,8}*64
8-fold quotients : {3,2,4}*48, {6,2,2}*48
12-fold quotients : {2,2,4}*32, {2,4,2}*32
16-fold quotients : {3,2,2}*24
24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,4,8}*768a, {6,8,8}*768a, {6,8,8}*768b, {12,4,8}*768a, {6,4,16}*768a, {6,4,16}*768b
3-fold covers : {18,4,8}*1152a, {6,12,8}*1152b, {6,12,8}*1152c, {6,4,24}*1152a
5-fold covers : {30,4,8}*1920a, {6,20,8}*1920a, {6,4,40}*1920a
Permutation Representation (GAP) :
```s0 := (  1, 25)(  2, 27)(  3, 26)(  4, 28)(  5, 30)(  6, 29)(  7, 31)(  8, 33)
(  9, 32)( 10, 34)( 11, 36)( 12, 35)( 13, 37)( 14, 39)( 15, 38)( 16, 40)
( 17, 42)( 18, 41)( 19, 43)( 20, 45)( 21, 44)( 22, 46)( 23, 48)( 24, 47)
( 49, 73)( 50, 75)( 51, 74)( 52, 76)( 53, 78)( 54, 77)( 55, 79)( 56, 81)
( 57, 80)( 58, 82)( 59, 84)( 60, 83)( 61, 85)( 62, 87)( 63, 86)( 64, 88)
( 65, 90)( 66, 89)( 67, 91)( 68, 93)( 69, 92)( 70, 94)( 71, 96)( 72, 95)
( 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)
(145,169)(146,171)(147,170)(148,172)(149,174)(150,173)(151,175)(152,177)
(153,176)(154,178)(155,180)(156,179)(157,181)(158,183)(159,182)(160,184)
(161,186)(162,185)(163,187)(164,189)(165,188)(166,190)(167,192)(168,191);;
s1 := (  1, 74)(  2, 73)(  3, 75)(  4, 77)(  5, 76)(  6, 78)(  7, 80)(  8, 79)
(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)
( 17, 88)( 18, 90)( 19, 92)( 20, 91)( 21, 93)( 22, 95)( 23, 94)( 24, 96)
( 25, 50)( 26, 49)( 27, 51)( 28, 53)( 29, 52)( 30, 54)( 31, 56)( 32, 55)
( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 62)( 38, 61)( 39, 63)( 40, 65)
( 41, 64)( 42, 66)( 43, 68)( 44, 67)( 45, 69)( 46, 71)( 47, 70)( 48, 72)
( 97,170)( 98,169)( 99,171)(100,173)(101,172)(102,174)(103,176)(104,175)
(105,177)(106,179)(107,178)(108,180)(109,182)(110,181)(111,183)(112,185)
(113,184)(114,186)(115,188)(116,187)(117,189)(118,191)(119,190)(120,192)
(121,146)(122,145)(123,147)(124,149)(125,148)(126,150)(127,152)(128,151)
(129,153)(130,155)(131,154)(132,156)(133,158)(134,157)(135,159)(136,161)
(137,160)(138,162)(139,164)(140,163)(141,165)(142,167)(143,166)(144,168);;
s2 := ( 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);;
s3 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)(  8,104)
(  9,105)( 10,106)( 11,107)( 12,108)( 13,112)( 14,113)( 15,114)( 16,109)
( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)( 24,117)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)( 32,128)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,136)( 38,137)( 39,138)( 40,133)
( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)( 56,152)
( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)( 64,157)
( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)( 72,165)
( 73,169)( 74,170)( 75,171)( 76,172)( 77,173)( 78,174)( 79,175)( 80,176)
( 81,177)( 82,178)( 83,179)( 84,180)( 85,184)( 86,185)( 87,186)( 88,181)
( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)( 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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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, 25)(  2, 27)(  3, 26)(  4, 28)(  5, 30)(  6, 29)(  7, 31)
(  8, 33)(  9, 32)( 10, 34)( 11, 36)( 12, 35)( 13, 37)( 14, 39)( 15, 38)
( 16, 40)( 17, 42)( 18, 41)( 19, 43)( 20, 45)( 21, 44)( 22, 46)( 23, 48)
( 24, 47)( 49, 73)( 50, 75)( 51, 74)( 52, 76)( 53, 78)( 54, 77)( 55, 79)
( 56, 81)( 57, 80)( 58, 82)( 59, 84)( 60, 83)( 61, 85)( 62, 87)( 63, 86)
( 64, 88)( 65, 90)( 66, 89)( 67, 91)( 68, 93)( 69, 92)( 70, 94)( 71, 96)
( 72, 95)( 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)(145,169)(146,171)(147,170)(148,172)(149,174)(150,173)(151,175)
(152,177)(153,176)(154,178)(155,180)(156,179)(157,181)(158,183)(159,182)
(160,184)(161,186)(162,185)(163,187)(164,189)(165,188)(166,190)(167,192)
(168,191);
s1 := Sym(192)!(  1, 74)(  2, 73)(  3, 75)(  4, 77)(  5, 76)(  6, 78)(  7, 80)
(  8, 79)(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13, 86)( 14, 85)( 15, 87)
( 16, 89)( 17, 88)( 18, 90)( 19, 92)( 20, 91)( 21, 93)( 22, 95)( 23, 94)
( 24, 96)( 25, 50)( 26, 49)( 27, 51)( 28, 53)( 29, 52)( 30, 54)( 31, 56)
( 32, 55)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 62)( 38, 61)( 39, 63)
( 40, 65)( 41, 64)( 42, 66)( 43, 68)( 44, 67)( 45, 69)( 46, 71)( 47, 70)
( 48, 72)( 97,170)( 98,169)( 99,171)(100,173)(101,172)(102,174)(103,176)
(104,175)(105,177)(106,179)(107,178)(108,180)(109,182)(110,181)(111,183)
(112,185)(113,184)(114,186)(115,188)(116,187)(117,189)(118,191)(119,190)
(120,192)(121,146)(122,145)(123,147)(124,149)(125,148)(126,150)(127,152)
(128,151)(129,153)(130,155)(131,154)(132,156)(133,158)(134,157)(135,159)
(136,161)(137,160)(138,162)(139,164)(140,163)(141,165)(142,167)(143,166)
(144,168);
s2 := 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);
s3 := Sym(192)!(  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)
(  8,104)(  9,105)( 10,106)( 11,107)( 12,108)( 13,112)( 14,113)( 15,114)
( 16,109)( 17,110)( 18,111)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)
( 24,117)( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)
( 32,128)( 33,129)( 34,130)( 35,131)( 36,132)( 37,136)( 38,137)( 39,138)
( 40,133)( 41,134)( 42,135)( 43,142)( 44,143)( 45,144)( 46,139)( 47,140)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,149)( 54,150)( 55,151)
( 56,152)( 57,153)( 58,154)( 59,155)( 60,156)( 61,160)( 62,161)( 63,162)
( 64,157)( 65,158)( 66,159)( 67,166)( 68,167)( 69,168)( 70,163)( 71,164)
( 72,165)( 73,169)( 74,170)( 75,171)( 76,172)( 77,173)( 78,174)( 79,175)
( 80,176)( 81,177)( 82,178)( 83,179)( 84,180)( 85,184)( 86,185)( 87,186)
( 88,181)( 89,182)( 90,183)( 91,190)( 92,191)( 93,192)( 94,187)( 95,188)
( 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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope