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# Polytope of Type {4,6,4}

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

```
References : None.
to this polytope