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# Polytope of Type {4,12,4}

Atlas Canonical Name : {4,12,4}*384a
Also Known As : {{4,12|2},{12,4|2}}. if this polytope has another name.
Group : SmallGroup(384,6428)
Rank : 4
Schlafli Type : {4,12,4}
Number of vertices, edges, etc : 4, 24, 24, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,12,4,2} of size 768
Vertex Figure Of :
{2,4,12,4} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,12,4}*192a, {4,12,2}*192a, {4,6,4}*192a
3-fold quotients : {4,4,4}*128
4-fold quotients : {2,12,2}*96, {2,6,4}*96a, {4,6,2}*96a
6-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
8-fold quotients : {2,6,2}*48
12-fold quotients : {2,2,4}*32, {2,4,2}*32, {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 : {4,12,8}*768a, {8,12,4}*768a, {4,12,8}*768b, {8,12,4}*768b, {4,24,4}*768a, {4,12,4}*768a, {4,12,4}*768b, {4,24,4}*768b, {4,24,4}*768c, {4,24,4}*768d
3-fold covers : {4,36,4}*1152a, {4,12,12}*1152a, {4,12,12}*1152b, {12,12,4}*1152a, {12,12,4}*1152b
5-fold covers : {4,60,4}*1920a, {4,12,20}*1920a, {20,12,4}*1920a
Permutation Representation (GAP) :
```s0 := (  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);;
s1 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 42)( 30, 41)( 31, 43)( 32, 45)
( 33, 44)( 34, 46)( 35, 48)( 36, 47)( 49, 55)( 50, 57)( 51, 56)( 52, 58)
( 53, 60)( 54, 59)( 61, 67)( 62, 69)( 63, 68)( 64, 70)( 65, 72)( 66, 71)
( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)( 79, 85)( 80, 87)
( 81, 86)( 82, 88)( 83, 90)( 84, 89)( 97,109)( 98,111)( 99,110)(100,112)
(101,114)(102,113)(103,115)(104,117)(105,116)(106,118)(107,120)(108,119)
(122,123)(125,126)(128,129)(131,132)(134,135)(137,138)(140,141)(143,144)
(145,163)(146,165)(147,164)(148,166)(149,168)(150,167)(151,157)(152,159)
(153,158)(154,160)(155,162)(156,161)(169,175)(170,177)(171,176)(172,178)
(173,180)(174,179)(181,187)(182,189)(183,188)(184,190)(185,192)(186,191);;
s2 := (  1, 50)(  2, 49)(  3, 51)(  4, 53)(  5, 52)(  6, 54)(  7, 56)(  8, 55)
(  9, 57)( 10, 59)( 11, 58)( 12, 60)( 13, 62)( 14, 61)( 15, 63)( 16, 65)
( 17, 64)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 71)( 23, 70)( 24, 72)
( 25, 74)( 26, 73)( 27, 75)( 28, 77)( 29, 76)( 30, 78)( 31, 80)( 32, 79)
( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 86)( 38, 85)( 39, 87)( 40, 89)
( 41, 88)( 42, 90)( 43, 92)( 44, 91)( 45, 93)( 46, 95)( 47, 94)( 48, 96)
( 97,146)( 98,145)( 99,147)(100,149)(101,148)(102,150)(103,152)(104,151)
(105,153)(106,155)(107,154)(108,156)(109,158)(110,157)(111,159)(112,161)
(113,160)(114,162)(115,164)(116,163)(117,165)(118,167)(119,166)(120,168)
(121,170)(122,169)(123,171)(124,173)(125,172)(126,174)(127,176)(128,175)
(129,177)(130,179)(131,178)(132,180)(133,182)(134,181)(135,183)(136,185)
(137,184)(138,186)(139,188)(140,187)(141,189)(142,191)(143,190)(144,192);;
s3 := (  1,121)(  2,122)(  3,123)(  4,124)(  5,125)(  6,126)(  7,127)(  8,128)
(  9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)
( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,172)( 50,173)( 51,174)( 52,169)( 53,170)( 54,171)( 55,178)( 56,179)
( 57,180)( 58,175)( 59,176)( 60,177)( 61,184)( 62,185)( 63,186)( 64,181)
( 65,182)( 66,183)( 67,190)( 68,191)( 69,192)( 70,187)( 71,188)( 72,189)
( 73,148)( 74,149)( 75,150)( 76,145)( 77,146)( 78,147)( 79,154)( 80,155)
( 81,156)( 82,151)( 83,152)( 84,153)( 85,160)( 86,161)( 87,162)( 88,157)
( 89,158)( 90,159)( 91,166)( 92,167)( 93,168)( 94,163)( 95,164)( 96,165);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope