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# Polytope of Type {4,24}

Atlas Canonical Name : {4,24}*768i
if this polytope has a name.
Group : SmallGroup(768,1087719)
Rank : 3
Schlafli Type : {4,24}
Number of vertices, edges, etc : 16, 192, 96
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12}*384d, {4,24}*384c, {4,24}*384d
4-fold quotients : {4,24}*192a, {4,24}*192c, {4,24}*192d, {4,12}*192b, {4,6}*192b, {4,12}*192c
8-fold quotients : {4,12}*96a, {2,24}*96, {4,12}*96b, {4,12}*96c, {4,6}*96
12-fold quotients : {4,8}*64a
16-fold quotients : {2,12}*48, {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {4,4}*32, {2,8}*32
32-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {2,4}*16, {4,2}*16
64-fold quotients : {2,3}*12
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)(104,149)
(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)(112,157)
(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)(120,165)
(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)(128,173)
(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)(136,181)
(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189);;
s1 := (  1,121)(  2,122)(  3,124)(  4,123)(  5,129)(  6,130)(  7,132)(  8,131)
(  9,125)( 10,126)( 11,128)( 12,127)( 13,133)( 14,134)( 15,136)( 16,135)
( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)( 24,139)
( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)( 32,107)
( 33,101)( 34,102)( 35,104)( 36,103)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)
( 49,169)( 50,170)( 51,172)( 52,171)( 53,177)( 54,178)( 55,180)( 56,179)
( 57,173)( 58,174)( 59,176)( 60,175)( 61,181)( 62,182)( 63,184)( 64,183)
( 65,189)( 66,190)( 67,192)( 68,191)( 69,185)( 70,186)( 71,188)( 72,187)
( 73,145)( 74,146)( 75,148)( 76,147)( 77,153)( 78,154)( 79,156)( 80,155)
( 81,149)( 82,150)( 83,152)( 84,151)( 85,157)( 86,158)( 87,160)( 88,159)
( 89,165)( 90,166)( 91,168)( 92,167)( 93,161)( 94,162)( 95,164)( 96,163);;
s2 := (  1,  9)(  2, 11)(  3, 10)(  4, 12)(  6,  7)( 13, 21)( 14, 23)( 15, 22)
( 16, 24)( 18, 19)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 41)( 30, 43)
( 31, 42)( 32, 44)( 33, 37)( 34, 39)( 35, 38)( 36, 40)( 49, 57)( 50, 59)
( 51, 58)( 52, 60)( 54, 55)( 61, 69)( 62, 71)( 63, 70)( 64, 72)( 66, 67)
( 73, 93)( 74, 95)( 75, 94)( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)
( 81, 85)( 82, 87)( 83, 86)( 84, 88)( 97,129)( 98,131)( 99,130)(100,132)
(101,125)(102,127)(103,126)(104,128)(105,121)(106,123)(107,122)(108,124)
(109,141)(110,143)(111,142)(112,144)(113,137)(114,139)(115,138)(116,140)
(117,133)(118,135)(119,134)(120,136)(145,177)(146,179)(147,178)(148,180)
(149,173)(150,175)(151,174)(152,176)(153,169)(154,171)(155,170)(156,172)
(157,189)(158,191)(159,190)(160,192)(161,185)(162,187)(163,186)(164,188)
(165,181)(166,183)(167,182)(168,184);;
poly := Group([s0,s1,s2]);;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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