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# Polytope of Type {8,44}

Atlas Canonical Name : {8,44}*704b
if this polytope has a name.
Group : SmallGroup(704,354)
Rank : 3
Schlafli Type : {8,44}
Number of vertices, edges, etc : 8, 176, 44
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 :
{8,44,2} of size 1408
Vertex Figure Of :
{2,8,44} 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 : {8,4}*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 : {8,44}*1408a, {8,88}*1408b, {8,88}*1408d
Permutation Representation (GAP) :
```s0 := ( 23, 34)( 24, 35)( 25, 36)( 26, 37)( 27, 38)( 28, 39)( 29, 40)( 30, 41)
( 31, 42)( 32, 43)( 33, 44)( 45, 56)( 46, 57)( 47, 58)( 48, 59)( 49, 60)
( 50, 61)( 51, 62)( 52, 63)( 53, 64)( 54, 65)( 55, 66)( 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)( 23,122)( 24,132)
( 25,131)( 26,130)( 27,129)( 28,128)( 29,127)( 30,126)( 31,125)( 32,124)
( 33,123)( 34,111)( 35,121)( 36,120)( 37,119)( 38,118)( 39,117)( 40,116)
( 41,115)( 42,114)( 43,113)( 44,112)( 45,133)( 46,143)( 47,142)( 48,141)
( 49,140)( 50,139)( 51,138)( 52,137)( 53,136)( 54,135)( 55,134)( 56,144)
( 57,154)( 58,153)( 59,152)( 60,151)( 61,150)( 62,149)( 63,148)( 64,147)
( 65,146)( 66,145)( 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, 46)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 67, 79)
( 68, 78)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73, 84)( 74, 83)( 75, 82)
( 76, 81)( 77, 80)( 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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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(176)!( 23, 34)( 24, 35)( 25, 36)( 26, 37)( 27, 38)( 28, 39)( 29, 40)
( 30, 41)( 31, 42)( 32, 43)( 33, 44)( 45, 56)( 46, 57)( 47, 58)( 48, 59)
( 49, 60)( 50, 61)( 51, 62)( 52, 63)( 53, 64)( 54, 65)( 55, 66)( 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)( 23,122)
( 24,132)( 25,131)( 26,130)( 27,129)( 28,128)( 29,127)( 30,126)( 31,125)
( 32,124)( 33,123)( 34,111)( 35,121)( 36,120)( 37,119)( 38,118)( 39,117)
( 40,116)( 41,115)( 42,114)( 43,113)( 44,112)( 45,133)( 46,143)( 47,142)
( 48,141)( 49,140)( 50,139)( 51,138)( 52,137)( 53,136)( 54,135)( 55,134)
( 56,144)( 57,154)( 58,153)( 59,152)( 60,151)( 61,150)( 62,149)( 63,148)
( 64,147)( 65,146)( 66,145)( 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, 46)( 47, 55)
( 48, 54)( 49, 53)( 50, 52)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)
( 67, 79)( 68, 78)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73, 84)( 74, 83)
( 75, 82)( 76, 81)( 77, 80)( 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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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