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# Polytope of Type {4,8,6}

Atlas Canonical Name : {4,8,6}*384b
if this polytope has a name.
Group : SmallGroup(384,12852)
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
3-fold quotients : {4,8,2}*128b
4-fold quotients : {2,4,6}*96a, {4,2,6}*96
6-fold quotients : {4,4,2}*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}*768c, {8,8,6}*768d, {4,8,12}*768b
3-fold covers : {4,8,18}*1152b, {12,8,6}*1152b, {4,24,6}*1152d, {4,24,6}*1152f
5-fold covers : {4,8,30}*1920b, {20,8,6}*1920b, {4,40,6}*1920b
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, 64)( 14, 65)( 15, 66)( 16, 61)
( 17, 62)( 18, 63)( 19, 70)( 20, 71)( 21, 72)( 22, 67)( 23, 68)( 24, 69)
( 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, 88)( 38, 89)( 39, 90)( 40, 85)
( 41, 86)( 42, 87)( 43, 94)( 44, 95)( 45, 96)( 46, 91)( 47, 92)( 48, 93)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,160)(110,161)(111,162)(112,157)
(113,158)(114,159)(115,166)(116,167)(117,168)(118,163)(119,164)(120,165)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,184)(134,185)(135,186)(136,181)
(137,182)(138,183)(139,190)(140,191)(141,192)(142,187)(143,188)(144,189);;
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)(145,166)(146,167)(147,168)(148,163)
(149,164)(150,165)(151,160)(152,161)(153,162)(154,157)(155,158)(156,159)
(169,190)(170,191)(171,192)(172,187)(173,188)(174,189)(175,184)(176,185)
(177,186)(178,181)(179,182)(180,183);;
s2 := (  1,121)(  2,123)(  3,122)(  4,124)(  5,126)(  6,125)(  7,130)(  8,132)
(  9,131)( 10,127)( 11,129)( 12,128)( 13,136)( 14,138)( 15,137)( 16,133)
( 17,135)( 18,134)( 19,139)( 20,141)( 21,140)( 22,142)( 23,144)( 24,143)
( 25, 97)( 26, 99)( 27, 98)( 28,100)( 29,102)( 30,101)( 31,106)( 32,108)
( 33,107)( 34,103)( 35,105)( 36,104)( 37,112)( 38,114)( 39,113)( 40,109)
( 41,111)( 42,110)( 43,115)( 44,117)( 45,116)( 46,118)( 47,120)( 48,119)
( 49,169)( 50,171)( 51,170)( 52,172)( 53,174)( 54,173)( 55,178)( 56,180)
( 57,179)( 58,175)( 59,177)( 60,176)( 61,184)( 62,186)( 63,185)( 64,181)
( 65,183)( 66,182)( 67,187)( 68,189)( 69,188)( 70,190)( 71,192)( 72,191)
( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,154)( 80,156)
( 81,155)( 82,151)( 83,153)( 84,152)( 85,160)( 86,162)( 87,161)( 88,157)
( 89,159)( 90,158)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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, 64)( 14, 65)( 15, 66)
( 16, 61)( 17, 62)( 18, 63)( 19, 70)( 20, 71)( 21, 72)( 22, 67)( 23, 68)
( 24, 69)( 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, 88)( 38, 89)( 39, 90)
( 40, 85)( 41, 86)( 42, 87)( 43, 94)( 44, 95)( 45, 96)( 46, 91)( 47, 92)
( 48, 93)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,160)(110,161)(111,162)
(112,157)(113,158)(114,159)(115,166)(116,167)(117,168)(118,163)(119,164)
(120,165)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,184)(134,185)(135,186)
(136,181)(137,182)(138,183)(139,190)(140,191)(141,192)(142,187)(143,188)
(144,189);
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)(145,166)(146,167)(147,168)
(148,163)(149,164)(150,165)(151,160)(152,161)(153,162)(154,157)(155,158)
(156,159)(169,190)(170,191)(171,192)(172,187)(173,188)(174,189)(175,184)
(176,185)(177,186)(178,181)(179,182)(180,183);
s2 := Sym(192)!(  1,121)(  2,123)(  3,122)(  4,124)(  5,126)(  6,125)(  7,130)
(  8,132)(  9,131)( 10,127)( 11,129)( 12,128)( 13,136)( 14,138)( 15,137)
( 16,133)( 17,135)( 18,134)( 19,139)( 20,141)( 21,140)( 22,142)( 23,144)
( 24,143)( 25, 97)( 26, 99)( 27, 98)( 28,100)( 29,102)( 30,101)( 31,106)
( 32,108)( 33,107)( 34,103)( 35,105)( 36,104)( 37,112)( 38,114)( 39,113)
( 40,109)( 41,111)( 42,110)( 43,115)( 44,117)( 45,116)( 46,118)( 47,120)
( 48,119)( 49,169)( 50,171)( 51,170)( 52,172)( 53,174)( 54,173)( 55,178)
( 56,180)( 57,179)( 58,175)( 59,177)( 60,176)( 61,184)( 62,186)( 63,185)
( 64,181)( 65,183)( 66,182)( 67,187)( 68,189)( 69,188)( 70,190)( 71,192)
( 72,191)( 73,145)( 74,147)( 75,146)( 76,148)( 77,150)( 78,149)( 79,154)
( 80,156)( 81,155)( 82,151)( 83,153)( 84,152)( 85,160)( 86,162)( 87,161)
( 88,157)( 89,159)( 90,158)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)
( 96,167);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope