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# Polytope of Type {8,24}

Atlas Canonical Name : {8,24}*384c
if this polytope has a name.
Group : SmallGroup(384,926)
Rank : 3
Schlafli Type : {8,24}
Number of vertices, edges, etc : 8, 96, 24
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,24,2} of size 768
Vertex Figure Of :
{2,8,24} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,24}*192b, {8,12}*192b
3-fold quotients : {8,8}*128d
4-fold quotients : {4,12}*96a
6-fold quotients : {4,8}*64b, {8,4}*64b
8-fold quotients : {2,12}*48, {4,6}*48a
12-fold quotients : {4,4}*32
16-fold quotients : {2,6}*24
24-fold quotients : {2,4}*16, {4,2}*16
32-fold quotients : {2,3}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,24}*768a
3-fold covers : {8,72}*1152d, {24,24}*1152j, {24,24}*1152k
5-fold covers : {8,120}*1920d, {40,24}*1920d
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, 70)( 14, 71)( 15, 72)( 16, 67)
( 17, 68)( 18, 69)( 19, 64)( 20, 65)( 21, 66)( 22, 61)( 23, 62)( 24, 63)
( 25, 76)( 26, 77)( 27, 78)( 28, 73)( 29, 74)( 30, 75)( 31, 82)( 32, 83)
( 33, 84)( 34, 79)( 35, 80)( 36, 81)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 85)( 44, 86)( 45, 87)( 46, 88)( 47, 89)( 48, 90)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,166)(110,167)(111,168)(112,163)
(113,164)(114,165)(115,160)(116,161)(117,162)(118,157)(119,158)(120,159)
(121,172)(122,173)(123,174)(124,169)(125,170)(126,171)(127,178)(128,179)
(129,180)(130,175)(131,176)(132,177)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186);;
s1 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 13, 22)( 14, 24)( 15, 23)( 16, 19)
( 17, 21)( 18, 20)( 25, 31)( 26, 33)( 27, 32)( 28, 34)( 29, 36)( 30, 35)
( 37, 40)( 38, 42)( 39, 41)( 43, 46)( 44, 48)( 45, 47)( 49, 61)( 50, 63)
( 51, 62)( 52, 64)( 53, 66)( 54, 65)( 55, 67)( 56, 69)( 57, 68)( 58, 70)
( 59, 72)( 60, 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)( 97,121)( 98,123)
( 99,122)(100,124)(101,126)(102,125)(103,127)(104,129)(105,128)(106,130)
(107,132)(108,131)(109,142)(110,144)(111,143)(112,139)(113,141)(114,140)
(115,136)(116,138)(117,137)(118,133)(119,135)(120,134)(145,184)(146,186)
(147,185)(148,181)(149,183)(150,182)(151,190)(152,192)(153,191)(154,187)
(155,189)(156,188)(157,172)(158,174)(159,173)(160,169)(161,171)(162,170)
(163,178)(164,180)(165,179)(166,175)(167,177)(168,176);;
s2 := (  1, 98)(  2, 97)(  3, 99)(  4,101)(  5,100)(  6,102)(  7,104)(  8,103)
(  9,105)( 10,107)( 11,106)( 12,108)( 13,113)( 14,112)( 15,114)( 16,110)
( 17,109)( 18,111)( 19,119)( 20,118)( 21,120)( 22,116)( 23,115)( 24,117)
( 25,128)( 26,127)( 27,129)( 28,131)( 29,130)( 30,132)( 31,122)( 32,121)
( 33,123)( 34,125)( 35,124)( 36,126)( 37,143)( 38,142)( 39,144)( 40,140)
( 41,139)( 42,141)( 43,137)( 44,136)( 45,138)( 46,134)( 47,133)( 48,135)
( 49,146)( 50,145)( 51,147)( 52,149)( 53,148)( 54,150)( 55,152)( 56,151)
( 57,153)( 58,155)( 59,154)( 60,156)( 61,161)( 62,160)( 63,162)( 64,158)
( 65,157)( 66,159)( 67,167)( 68,166)( 69,168)( 70,164)( 71,163)( 72,165)
( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,170)( 80,169)
( 81,171)( 82,173)( 83,172)( 84,174)( 85,191)( 86,190)( 87,192)( 88,188)
( 89,187)( 90,189)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*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, 70)( 14, 71)( 15, 72)
( 16, 67)( 17, 68)( 18, 69)( 19, 64)( 20, 65)( 21, 66)( 22, 61)( 23, 62)
( 24, 63)( 25, 76)( 26, 77)( 27, 78)( 28, 73)( 29, 74)( 30, 75)( 31, 82)
( 32, 83)( 33, 84)( 34, 79)( 35, 80)( 36, 81)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 85)( 44, 86)( 45, 87)( 46, 88)( 47, 89)
( 48, 90)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,166)(110,167)(111,168)
(112,163)(113,164)(114,165)(115,160)(116,161)(117,162)(118,157)(119,158)
(120,159)(121,172)(122,173)(123,174)(124,169)(125,170)(126,171)(127,178)
(128,179)(129,180)(130,175)(131,176)(132,177)(133,187)(134,188)(135,189)
(136,190)(137,191)(138,192)(139,181)(140,182)(141,183)(142,184)(143,185)
(144,186);
s1 := Sym(192)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 13, 22)( 14, 24)( 15, 23)
( 16, 19)( 17, 21)( 18, 20)( 25, 31)( 26, 33)( 27, 32)( 28, 34)( 29, 36)
( 30, 35)( 37, 40)( 38, 42)( 39, 41)( 43, 46)( 44, 48)( 45, 47)( 49, 61)
( 50, 63)( 51, 62)( 52, 64)( 53, 66)( 54, 65)( 55, 67)( 56, 69)( 57, 68)
( 58, 70)( 59, 72)( 60, 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)( 97,121)
( 98,123)( 99,122)(100,124)(101,126)(102,125)(103,127)(104,129)(105,128)
(106,130)(107,132)(108,131)(109,142)(110,144)(111,143)(112,139)(113,141)
(114,140)(115,136)(116,138)(117,137)(118,133)(119,135)(120,134)(145,184)
(146,186)(147,185)(148,181)(149,183)(150,182)(151,190)(152,192)(153,191)
(154,187)(155,189)(156,188)(157,172)(158,174)(159,173)(160,169)(161,171)
(162,170)(163,178)(164,180)(165,179)(166,175)(167,177)(168,176);
s2 := Sym(192)!(  1, 98)(  2, 97)(  3, 99)(  4,101)(  5,100)(  6,102)(  7,104)
(  8,103)(  9,105)( 10,107)( 11,106)( 12,108)( 13,113)( 14,112)( 15,114)
( 16,110)( 17,109)( 18,111)( 19,119)( 20,118)( 21,120)( 22,116)( 23,115)
( 24,117)( 25,128)( 26,127)( 27,129)( 28,131)( 29,130)( 30,132)( 31,122)
( 32,121)( 33,123)( 34,125)( 35,124)( 36,126)( 37,143)( 38,142)( 39,144)
( 40,140)( 41,139)( 42,141)( 43,137)( 44,136)( 45,138)( 46,134)( 47,133)
( 48,135)( 49,146)( 50,145)( 51,147)( 52,149)( 53,148)( 54,150)( 55,152)
( 56,151)( 57,153)( 58,155)( 59,154)( 60,156)( 61,161)( 62,160)( 63,162)
( 64,158)( 65,157)( 66,159)( 67,167)( 68,166)( 69,168)( 70,164)( 71,163)
( 72,165)( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,170)
( 80,169)( 81,171)( 82,173)( 83,172)( 84,174)( 85,191)( 86,190)( 87,192)
( 88,188)( 89,187)( 90,189)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)
( 96,183);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2 >;

```
References : None.
to this polytope