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# Polytope of Type {44,4}

Atlas Canonical Name : {44,4}*704
if this polytope has a name.
Group : SmallGroup(704,273)
Rank : 3
Schlafli Type : {44,4}
Number of vertices, edges, etc : 88, 176, 8
Order of s0s1s2 : 44
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{44,4,2} of size 1408
Vertex Figure Of :
{2,44,4} of size 1408
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {44,4}*352
4-fold quotients : {44,2}*176, {22,4}*176
8-fold quotients : {22,2}*88
11-fold quotients : {4,4}*64
16-fold quotients : {11,2}*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 : {44,8}*1408a, {88,4}*1408a, {44,4}*1408, {88,4}*1408b, {44,8}*1408b
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 13, 22)( 14, 21)( 15, 20)
( 16, 19)( 17, 18)( 23, 34)( 24, 44)( 25, 43)( 26, 42)( 27, 41)( 28, 40)
( 29, 39)( 30, 38)( 31, 37)( 32, 36)( 33, 35)( 46, 55)( 47, 54)( 48, 53)
( 49, 52)( 50, 51)( 57, 66)( 58, 65)( 59, 64)( 60, 63)( 61, 62)( 67, 78)
( 68, 88)( 69, 87)( 70, 86)( 71, 85)( 72, 84)( 73, 83)( 74, 82)( 75, 81)
( 76, 80)( 77, 79)( 89,133)( 90,143)( 91,142)( 92,141)( 93,140)( 94,139)
( 95,138)( 96,137)( 97,136)( 98,135)( 99,134)(100,144)(101,154)(102,153)
(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)
(111,166)(112,176)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)
(119,169)(120,168)(121,167)(122,155)(123,165)(124,164)(125,163)(126,162)
(127,161)(128,160)(129,159)(130,158)(131,157)(132,156);;
s1 := (  1, 90)(  2, 89)(  3, 99)(  4, 98)(  5, 97)(  6, 96)(  7, 95)(  8, 94)
(  9, 93)( 10, 92)( 11, 91)( 12,101)( 13,100)( 14,110)( 15,109)( 16,108)
( 17,107)( 18,106)( 19,105)( 20,104)( 21,103)( 22,102)( 23,112)( 24,111)
( 25,121)( 26,120)( 27,119)( 28,118)( 29,117)( 30,116)( 31,115)( 32,114)
( 33,113)( 34,123)( 35,122)( 36,132)( 37,131)( 38,130)( 39,129)( 40,128)
( 41,127)( 42,126)( 43,125)( 44,124)( 45,134)( 46,133)( 47,143)( 48,142)
( 49,141)( 50,140)( 51,139)( 52,138)( 53,137)( 54,136)( 55,135)( 56,145)
( 57,144)( 58,154)( 59,153)( 60,152)( 61,151)( 62,150)( 63,149)( 64,148)
( 65,147)( 66,146)( 67,156)( 68,155)( 69,165)( 70,164)( 71,163)( 72,162)
( 73,161)( 74,160)( 75,159)( 76,158)( 77,157)( 78,167)( 79,166)( 80,176)
( 81,175)( 82,174)( 83,173)( 84,172)( 85,171)( 86,170)( 87,169)( 88,168);;
s2 := ( 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);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope