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# Polytope of Type {28,18}

Atlas Canonical Name : {28,18}*1008b
if this polytope has a name.
Group : SmallGroup(1008,500)
Rank : 3
Schlafli Type : {28,18}
Number of vertices, edges, etc : 28, 252, 18
Order of s0s1s2 : 63
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {28,6}*336b
7-fold quotients : {4,18}*144c
14-fold quotients : {4,9}*72
21-fold quotients : {4,6}*48b
42-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 75)( 14, 76)
( 15, 73)( 16, 74)( 17, 79)( 18, 80)( 19, 77)( 20, 78)( 21, 83)( 22, 84)
( 23, 81)( 24, 82)( 25, 63)( 26, 64)( 27, 61)( 28, 62)( 29, 67)( 30, 68)
( 31, 65)( 32, 66)( 33, 71)( 34, 72)( 35, 69)( 36, 70)( 37, 51)( 38, 52)
( 39, 49)( 40, 50)( 41, 55)( 42, 56)( 43, 53)( 44, 54)( 45, 59)( 46, 60)
( 47, 57)( 48, 58)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97,159)( 98,160)( 99,157)(100,158)(101,163)(102,164)(103,161)(104,162)
(105,167)(106,168)(107,165)(108,166)(109,147)(110,148)(111,145)(112,146)
(113,151)(114,152)(115,149)(116,150)(117,155)(118,156)(119,153)(120,154)
(121,135)(122,136)(123,133)(124,134)(125,139)(126,140)(127,137)(128,138)
(129,143)(130,144)(131,141)(132,142)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,243)(182,244)(183,241)(184,242)(185,247)(186,248)
(187,245)(188,246)(189,251)(190,252)(191,249)(192,250)(193,231)(194,232)
(195,229)(196,230)(197,235)(198,236)(199,233)(200,234)(201,239)(202,240)
(203,237)(204,238)(205,219)(206,220)(207,217)(208,218)(209,223)(210,224)
(211,221)(212,222)(213,227)(214,228)(215,225)(216,226);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 73)( 26, 74)( 27, 76)( 28, 75)
( 29, 81)( 30, 82)( 31, 84)( 32, 83)( 33, 77)( 34, 78)( 35, 80)( 36, 79)
( 37, 61)( 38, 62)( 39, 64)( 40, 63)( 41, 69)( 42, 70)( 43, 72)( 44, 71)
( 45, 65)( 46, 66)( 47, 68)( 48, 67)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 85,189)( 86,190)( 87,192)( 88,191)( 89,185)( 90,186)( 91,188)
( 92,187)( 93,181)( 94,182)( 95,184)( 96,183)( 97,177)( 98,178)( 99,180)
(100,179)(101,173)(102,174)(103,176)(104,175)(105,169)(106,170)(107,172)
(108,171)(109,249)(110,250)(111,252)(112,251)(113,245)(114,246)(115,248)
(116,247)(117,241)(118,242)(119,244)(120,243)(121,237)(122,238)(123,240)
(124,239)(125,233)(126,234)(127,236)(128,235)(129,229)(130,230)(131,232)
(132,231)(133,225)(134,226)(135,228)(136,227)(137,221)(138,222)(139,224)
(140,223)(141,217)(142,218)(143,220)(144,219)(145,213)(146,214)(147,216)
(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)(155,208)
(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)(163,200)
(164,199)(165,193)(166,194)(167,196)(168,195);;
s2 := (  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 93)(  6, 96)(  7, 95)(  8, 94)
(  9, 89)( 10, 92)( 11, 91)( 12, 90)( 13, 97)( 14,100)( 15, 99)( 16, 98)
( 17,105)( 18,108)( 19,107)( 20,106)( 21,101)( 22,104)( 23,103)( 24,102)
( 25,109)( 26,112)( 27,111)( 28,110)( 29,117)( 30,120)( 31,119)( 32,118)
( 33,113)( 34,116)( 35,115)( 36,114)( 37,121)( 38,124)( 39,123)( 40,122)
( 41,129)( 42,132)( 43,131)( 44,130)( 45,125)( 46,128)( 47,127)( 48,126)
( 49,133)( 50,136)( 51,135)( 52,134)( 53,141)( 54,144)( 55,143)( 56,142)
( 57,137)( 58,140)( 59,139)( 60,138)( 61,145)( 62,148)( 63,147)( 64,146)
( 65,153)( 66,156)( 67,155)( 68,154)( 69,149)( 70,152)( 71,151)( 72,150)
( 73,157)( 74,160)( 75,159)( 76,158)( 77,165)( 78,168)( 79,167)( 80,166)
( 81,161)( 82,164)( 83,163)( 84,162)(169,177)(170,180)(171,179)(172,178)
(174,176)(181,189)(182,192)(183,191)(184,190)(186,188)(193,201)(194,204)
(195,203)(196,202)(198,200)(205,213)(206,216)(207,215)(208,214)(210,212)
(217,225)(218,228)(219,227)(220,226)(222,224)(229,237)(230,240)(231,239)
(232,238)(234,236)(241,249)(242,252)(243,251)(244,250)(246,248);;
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(252)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 75)
( 14, 76)( 15, 73)( 16, 74)( 17, 79)( 18, 80)( 19, 77)( 20, 78)( 21, 83)
( 22, 84)( 23, 81)( 24, 82)( 25, 63)( 26, 64)( 27, 61)( 28, 62)( 29, 67)
( 30, 68)( 31, 65)( 32, 66)( 33, 71)( 34, 72)( 35, 69)( 36, 70)( 37, 51)
( 38, 52)( 39, 49)( 40, 50)( 41, 55)( 42, 56)( 43, 53)( 44, 54)( 45, 59)
( 46, 60)( 47, 57)( 48, 58)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97,159)( 98,160)( 99,157)(100,158)(101,163)(102,164)(103,161)
(104,162)(105,167)(106,168)(107,165)(108,166)(109,147)(110,148)(111,145)
(112,146)(113,151)(114,152)(115,149)(116,150)(117,155)(118,156)(119,153)
(120,154)(121,135)(122,136)(123,133)(124,134)(125,139)(126,140)(127,137)
(128,138)(129,143)(130,144)(131,141)(132,142)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,243)(182,244)(183,241)(184,242)(185,247)
(186,248)(187,245)(188,246)(189,251)(190,252)(191,249)(192,250)(193,231)
(194,232)(195,229)(196,230)(197,235)(198,236)(199,233)(200,234)(201,239)
(202,240)(203,237)(204,238)(205,219)(206,220)(207,217)(208,218)(209,223)
(210,224)(211,221)(212,222)(213,227)(214,228)(215,225)(216,226);
s1 := Sym(252)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)
(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 73)( 26, 74)( 27, 76)
( 28, 75)( 29, 81)( 30, 82)( 31, 84)( 32, 83)( 33, 77)( 34, 78)( 35, 80)
( 36, 79)( 37, 61)( 38, 62)( 39, 64)( 40, 63)( 41, 69)( 42, 70)( 43, 72)
( 44, 71)( 45, 65)( 46, 66)( 47, 68)( 48, 67)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 85,189)( 86,190)( 87,192)( 88,191)( 89,185)( 90,186)
( 91,188)( 92,187)( 93,181)( 94,182)( 95,184)( 96,183)( 97,177)( 98,178)
( 99,180)(100,179)(101,173)(102,174)(103,176)(104,175)(105,169)(106,170)
(107,172)(108,171)(109,249)(110,250)(111,252)(112,251)(113,245)(114,246)
(115,248)(116,247)(117,241)(118,242)(119,244)(120,243)(121,237)(122,238)
(123,240)(124,239)(125,233)(126,234)(127,236)(128,235)(129,229)(130,230)
(131,232)(132,231)(133,225)(134,226)(135,228)(136,227)(137,221)(138,222)
(139,224)(140,223)(141,217)(142,218)(143,220)(144,219)(145,213)(146,214)
(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)
(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)
(163,200)(164,199)(165,193)(166,194)(167,196)(168,195);
s2 := Sym(252)!(  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 93)(  6, 96)(  7, 95)
(  8, 94)(  9, 89)( 10, 92)( 11, 91)( 12, 90)( 13, 97)( 14,100)( 15, 99)
( 16, 98)( 17,105)( 18,108)( 19,107)( 20,106)( 21,101)( 22,104)( 23,103)
( 24,102)( 25,109)( 26,112)( 27,111)( 28,110)( 29,117)( 30,120)( 31,119)
( 32,118)( 33,113)( 34,116)( 35,115)( 36,114)( 37,121)( 38,124)( 39,123)
( 40,122)( 41,129)( 42,132)( 43,131)( 44,130)( 45,125)( 46,128)( 47,127)
( 48,126)( 49,133)( 50,136)( 51,135)( 52,134)( 53,141)( 54,144)( 55,143)
( 56,142)( 57,137)( 58,140)( 59,139)( 60,138)( 61,145)( 62,148)( 63,147)
( 64,146)( 65,153)( 66,156)( 67,155)( 68,154)( 69,149)( 70,152)( 71,151)
( 72,150)( 73,157)( 74,160)( 75,159)( 76,158)( 77,165)( 78,168)( 79,167)
( 80,166)( 81,161)( 82,164)( 83,163)( 84,162)(169,177)(170,180)(171,179)
(172,178)(174,176)(181,189)(182,192)(183,191)(184,190)(186,188)(193,201)
(194,204)(195,203)(196,202)(198,200)(205,213)(206,216)(207,215)(208,214)
(210,212)(217,225)(218,228)(219,227)(220,226)(222,224)(229,237)(230,240)
(231,239)(232,238)(234,236)(241,249)(242,252)(243,251)(244,250)(246,248);
poly := sub<Sym(252)|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*s2*s1*s0*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope