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# Polytope of Type {4,6,4}

Atlas Canonical Name : {4,6,4}*384b
if this polytope has a name.
Group : SmallGroup(384,20051)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 8, 24, 24, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,4,2} of size 768
Vertex Figure Of :
{2,4,6,4} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,4}*192c, {4,6,2}*192
4-fold quotients : {2,6,4}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
8-fold quotients : {4,3,2}*48, {2,6,2}*48
12-fold quotients : {2,2,4}*32
16-fold quotients : {2,3,2}*24
24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,4}*768f, {4,6,4}*768d, {4,12,4}*768h, {4,6,8}*768a, {8,6,4}*768b, {8,6,4}*768c
3-fold covers : {4,18,4}*1152b, {4,6,12}*1152a, {12,6,4}*1152b, {12,6,4}*1152c, {4,6,12}*1152d
5-fold covers : {4,6,20}*1920a, {20,6,4}*1920b, {4,30,4}*1920b
Permutation Representation (GAP) :
```s0 := (  1, 99)(  2,100)(  3, 97)(  4, 98)(  5,103)(  6,104)(  7,101)(  8,102)
(  9,107)( 10,108)( 11,105)( 12,106)( 13,111)( 14,112)( 15,109)( 16,110)
( 17,115)( 18,116)( 19,113)( 20,114)( 21,119)( 22,120)( 23,117)( 24,118)
( 25,123)( 26,124)( 27,121)( 28,122)( 29,127)( 30,128)( 31,125)( 32,126)
( 33,131)( 34,132)( 35,129)( 36,130)( 37,135)( 38,136)( 39,133)( 40,134)
( 41,139)( 42,140)( 43,137)( 44,138)( 45,143)( 46,144)( 47,141)( 48,142)
( 49,147)( 50,148)( 51,145)( 52,146)( 53,151)( 54,152)( 55,149)( 56,150)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,159)( 62,160)( 63,157)( 64,158)
( 65,163)( 66,164)( 67,161)( 68,162)( 69,167)( 70,168)( 71,165)( 72,166)
( 73,171)( 74,172)( 75,169)( 76,170)( 77,175)( 78,176)( 79,173)( 80,174)
( 81,179)( 82,180)( 83,177)( 84,178)( 85,183)( 86,184)( 87,181)( 88,182)
( 89,187)( 90,188)( 91,185)( 92,186)( 93,191)( 94,192)( 95,189)( 96,190);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 21)( 18, 23)
( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 38, 39)
( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 50, 51)( 53, 57)( 54, 59)( 55, 58)
( 56, 60)( 62, 63)( 65, 69)( 66, 71)( 67, 70)( 68, 72)( 74, 75)( 77, 81)
( 78, 83)( 79, 82)( 80, 84)( 86, 87)( 89, 93)( 90, 95)( 91, 94)( 92, 96)
( 98, 99)(101,105)(102,107)(103,106)(104,108)(110,111)(113,117)(114,119)
(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)
(137,141)(138,143)(139,142)(140,144)(146,147)(149,153)(150,155)(151,154)
(152,156)(158,159)(161,165)(162,167)(163,166)(164,168)(170,171)(173,177)
(174,179)(175,178)(176,180)(182,183)(185,189)(186,191)(187,190)(188,192);;
s2 := (  1, 57)(  2, 60)(  3, 59)(  4, 58)(  5, 53)(  6, 56)(  7, 55)(  8, 54)
(  9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)( 24, 62)
( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 89)( 30, 92)( 31, 91)( 32, 90)
( 33, 85)( 34, 88)( 35, 87)( 36, 86)( 37, 81)( 38, 84)( 39, 83)( 40, 82)
( 41, 77)( 42, 80)( 43, 79)( 44, 78)( 45, 73)( 46, 76)( 47, 75)( 48, 74)
( 97,153)( 98,156)( 99,155)(100,154)(101,149)(102,152)(103,151)(104,150)
(105,145)(106,148)(107,147)(108,146)(109,165)(110,168)(111,167)(112,166)
(113,161)(114,164)(115,163)(116,162)(117,157)(118,160)(119,159)(120,158)
(121,189)(122,192)(123,191)(124,190)(125,185)(126,188)(127,187)(128,186)
(129,181)(130,184)(131,183)(132,182)(133,177)(134,180)(135,179)(136,178)
(137,173)(138,176)(139,175)(140,174)(141,169)(142,172)(143,171)(144,170);;
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, 37)( 14, 38)( 15, 39)( 16, 40)
( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)( 24, 48)
( 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, 85)( 62, 86)( 63, 87)( 64, 88)
( 65, 89)( 66, 90)( 67, 91)( 68, 92)( 69, 93)( 70, 94)( 71, 95)( 72, 96)
( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)
(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)
(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)
(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)
(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)
(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope