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Polytope of Type {4,88}

Atlas Canonical Name : {4,88}*704b
if this polytope has a name.
Group : SmallGroup(704,279)
Rank : 3
Schlafli Type : {4,88}
Number of vertices, edges, etc : 4, 176, 88
Order of s0s1s2 : 88
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,88,2} of size 1408
Vertex Figure Of :
{2,4,88} of size 1408
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,44}*352
4-fold quotients : {2,44}*176, {4,22}*176
8-fold quotients : {2,22}*88
11-fold quotients : {4,8}*64b
16-fold quotients : {2,11}*44
22-fold quotients : {4,4}*32
44-fold quotients : {2,4}*16, {4,2}*16
88-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,88}*1408a, {8,88}*1408a, {8,88}*1408d
Permutation Representation (GAP) :
```s0 := ( 45, 56)( 46, 57)( 47, 58)( 48, 59)( 49, 60)( 50, 61)( 51, 62)( 52, 63)
( 53, 64)( 54, 65)( 55, 66)( 67, 78)( 68, 79)( 69, 80)( 70, 81)( 71, 82)
( 72, 83)( 73, 84)( 74, 85)( 75, 86)( 76, 87)( 77, 88)( 89,111)( 90,112)
( 91,113)( 92,114)( 93,115)( 94,116)( 95,117)( 96,118)( 97,119)( 98,120)
( 99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)
(107,129)(108,130)(109,131)(110,132)(133,166)(134,167)(135,168)(136,169)
(137,170)(138,171)(139,172)(140,173)(141,174)(142,175)(143,176)(144,155)
(145,156)(146,157)(147,158)(148,159)(149,160)(150,161)(151,162)(152,163)
(153,164)(154,165);;
s1 := (  1, 89)(  2, 99)(  3, 98)(  4, 97)(  5, 96)(  6, 95)(  7, 94)(  8, 93)
(  9, 92)( 10, 91)( 11, 90)( 12,100)( 13,110)( 14,109)( 15,108)( 16,107)
( 17,106)( 18,105)( 19,104)( 20,103)( 21,102)( 22,101)( 23,111)( 24,121)
( 25,120)( 26,119)( 27,118)( 28,117)( 29,116)( 30,115)( 31,114)( 32,113)
( 33,112)( 34,122)( 35,132)( 36,131)( 37,130)( 38,129)( 39,128)( 40,127)
( 41,126)( 42,125)( 43,124)( 44,123)( 45,144)( 46,154)( 47,153)( 48,152)
( 49,151)( 50,150)( 51,149)( 52,148)( 53,147)( 54,146)( 55,145)( 56,133)
( 57,143)( 58,142)( 59,141)( 60,140)( 61,139)( 62,138)( 63,137)( 64,136)
( 65,135)( 66,134)( 67,166)( 68,176)( 69,175)( 70,174)( 71,173)( 72,172)
( 73,171)( 74,170)( 75,169)( 76,168)( 77,167)( 78,155)( 79,165)( 80,164)
( 81,163)( 82,162)( 83,161)( 84,160)( 85,159)( 86,158)( 87,157)( 88,156);;
s2 := (  1,  2)(  3, 11)(  4, 10)(  5,  9)(  6,  8)( 12, 13)( 14, 22)( 15, 21)
( 16, 20)( 17, 19)( 23, 35)( 24, 34)( 25, 44)( 26, 43)( 27, 42)( 28, 41)
( 29, 40)( 30, 39)( 31, 38)( 32, 37)( 33, 36)( 45, 57)( 46, 56)( 47, 66)
( 48, 65)( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 60)( 54, 59)( 55, 58)
( 67, 68)( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 78, 79)( 80, 88)( 81, 87)
( 82, 86)( 83, 85)( 89,134)( 90,133)( 91,143)( 92,142)( 93,141)( 94,140)
( 95,139)( 96,138)( 97,137)( 98,136)( 99,135)(100,145)(101,144)(102,154)
(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)
(111,167)(112,166)(113,176)(114,175)(115,174)(116,173)(117,172)(118,171)
(119,170)(120,169)(121,168)(122,156)(123,155)(124,165)(125,164)(126,163)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*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*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope