# Polytope of Type {4,124}

Atlas Canonical Name : {4,124}*992
Also Known As : {4,124|2}. if this polytope has another name.
Group : SmallGroup(992,77)
Rank : 3
Schlafli Type : {4,124}
Number of vertices, edges, etc : 4, 248, 124
Order of s0s1s2 : 124
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,124,2} of size 1984
Vertex Figure Of :
{2,4,124} of size 1984
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,124}*496, {4,62}*496
4-fold quotients : {2,62}*248
8-fold quotients : {2,31}*124
31-fold quotients : {4,4}*32
62-fold quotients : {2,4}*16, {4,2}*16
124-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,124}*1984a, {4,248}*1984a, {8,124}*1984b, {4,248}*1984b, {4,124}*1984
Permutation Representation (GAP) :
```s0 := (125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)
(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)
(141,172)(142,173)(143,174)(144,175)(145,176)(146,177)(147,178)(148,179)
(149,180)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(187,218)
(188,219)(189,220)(190,221)(191,222)(192,223)(193,224)(194,225)(195,226)
(196,227)(197,228)(198,229)(199,230)(200,231)(201,232)(202,233)(203,234)
(204,235)(205,236)(206,237)(207,238)(208,239)(209,240)(210,241)(211,242)
(212,243)(213,244)(214,245)(215,246)(216,247)(217,248);;
s1 := (  1,125)(  2,155)(  3,154)(  4,153)(  5,152)(  6,151)(  7,150)(  8,149)
(  9,148)( 10,147)( 11,146)( 12,145)( 13,144)( 14,143)( 15,142)( 16,141)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,136)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,131)( 27,130)( 28,129)( 29,128)( 30,127)( 31,126)( 32,156)
( 33,186)( 34,185)( 35,184)( 36,183)( 37,182)( 38,181)( 39,180)( 40,179)
( 41,178)( 42,177)( 43,176)( 44,175)( 45,174)( 46,173)( 47,172)( 48,171)
( 49,170)( 50,169)( 51,168)( 52,167)( 53,166)( 54,165)( 55,164)( 56,163)
( 57,162)( 58,161)( 59,160)( 60,159)( 61,158)( 62,157)( 63,187)( 64,217)
( 65,216)( 66,215)( 67,214)( 68,213)( 69,212)( 70,211)( 71,210)( 72,209)
( 73,208)( 74,207)( 75,206)( 76,205)( 77,204)( 78,203)( 79,202)( 80,201)
( 81,200)( 82,199)( 83,198)( 84,197)( 85,196)( 86,195)( 87,194)( 88,193)
( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,218)( 95,248)( 96,247)
( 97,246)( 98,245)( 99,244)(100,243)(101,242)(102,241)(103,240)(104,239)
(105,238)(106,237)(107,236)(108,235)(109,234)(110,233)(111,232)(112,231)
(113,230)(114,229)(115,228)(116,227)(117,226)(118,225)(119,224)(120,223)
(121,222)(122,221)(123,220)(124,219);;
s2 := (  1,  2)(  3, 31)(  4, 30)(  5, 29)(  6, 28)(  7, 27)(  8, 26)(  9, 25)
( 10, 24)( 11, 23)( 12, 22)( 13, 21)( 14, 20)( 15, 19)( 16, 18)( 32, 33)
( 34, 62)( 35, 61)( 36, 60)( 37, 59)( 38, 58)( 39, 57)( 40, 56)( 41, 55)
( 42, 54)( 43, 53)( 44, 52)( 45, 51)( 46, 50)( 47, 49)( 63, 64)( 65, 93)
( 66, 92)( 67, 91)( 68, 90)( 69, 89)( 70, 88)( 71, 87)( 72, 86)( 73, 85)
( 74, 84)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 94, 95)( 96,124)( 97,123)
( 98,122)( 99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)
(106,114)(107,113)(108,112)(109,111)(125,188)(126,187)(127,217)(128,216)
(129,215)(130,214)(131,213)(132,212)(133,211)(134,210)(135,209)(136,208)
(137,207)(138,206)(139,205)(140,204)(141,203)(142,202)(143,201)(144,200)
(145,199)(146,198)(147,197)(148,196)(149,195)(150,194)(151,193)(152,192)
(153,191)(154,190)(155,189)(156,219)(157,218)(158,248)(159,247)(160,246)
(161,245)(162,244)(163,243)(164,242)(165,241)(166,240)(167,239)(168,238)
(169,237)(170,236)(171,235)(172,234)(173,233)(174,232)(175,231)(176,230)
(177,229)(178,228)(179,227)(180,226)(181,225)(182,224)(183,223)(184,222)
(185,221)(186,220);;
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,
s0*s1*s2*s1*s0*s1*s2*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*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*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(248)!(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)
(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)
(140,171)(141,172)(142,173)(143,174)(144,175)(145,176)(146,177)(147,178)
(148,179)(149,180)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)
(187,218)(188,219)(189,220)(190,221)(191,222)(192,223)(193,224)(194,225)
(195,226)(196,227)(197,228)(198,229)(199,230)(200,231)(201,232)(202,233)
(203,234)(204,235)(205,236)(206,237)(207,238)(208,239)(209,240)(210,241)
(211,242)(212,243)(213,244)(214,245)(215,246)(216,247)(217,248);
s1 := Sym(248)!(  1,125)(  2,155)(  3,154)(  4,153)(  5,152)(  6,151)(  7,150)
(  8,149)(  9,148)( 10,147)( 11,146)( 12,145)( 13,144)( 14,143)( 15,142)
( 16,141)( 17,140)( 18,139)( 19,138)( 20,137)( 21,136)( 22,135)( 23,134)
( 24,133)( 25,132)( 26,131)( 27,130)( 28,129)( 29,128)( 30,127)( 31,126)
( 32,156)( 33,186)( 34,185)( 35,184)( 36,183)( 37,182)( 38,181)( 39,180)
( 40,179)( 41,178)( 42,177)( 43,176)( 44,175)( 45,174)( 46,173)( 47,172)
( 48,171)( 49,170)( 50,169)( 51,168)( 52,167)( 53,166)( 54,165)( 55,164)
( 56,163)( 57,162)( 58,161)( 59,160)( 60,159)( 61,158)( 62,157)( 63,187)
( 64,217)( 65,216)( 66,215)( 67,214)( 68,213)( 69,212)( 70,211)( 71,210)
( 72,209)( 73,208)( 74,207)( 75,206)( 76,205)( 77,204)( 78,203)( 79,202)
( 80,201)( 81,200)( 82,199)( 83,198)( 84,197)( 85,196)( 86,195)( 87,194)
( 88,193)( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,218)( 95,248)
( 96,247)( 97,246)( 98,245)( 99,244)(100,243)(101,242)(102,241)(103,240)
(104,239)(105,238)(106,237)(107,236)(108,235)(109,234)(110,233)(111,232)
(112,231)(113,230)(114,229)(115,228)(116,227)(117,226)(118,225)(119,224)
(120,223)(121,222)(122,221)(123,220)(124,219);
s2 := Sym(248)!(  1,  2)(  3, 31)(  4, 30)(  5, 29)(  6, 28)(  7, 27)(  8, 26)
(  9, 25)( 10, 24)( 11, 23)( 12, 22)( 13, 21)( 14, 20)( 15, 19)( 16, 18)
( 32, 33)( 34, 62)( 35, 61)( 36, 60)( 37, 59)( 38, 58)( 39, 57)( 40, 56)
( 41, 55)( 42, 54)( 43, 53)( 44, 52)( 45, 51)( 46, 50)( 47, 49)( 63, 64)
( 65, 93)( 66, 92)( 67, 91)( 68, 90)( 69, 89)( 70, 88)( 71, 87)( 72, 86)
( 73, 85)( 74, 84)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 94, 95)( 96,124)
( 97,123)( 98,122)( 99,121)(100,120)(101,119)(102,118)(103,117)(104,116)
(105,115)(106,114)(107,113)(108,112)(109,111)(125,188)(126,187)(127,217)
(128,216)(129,215)(130,214)(131,213)(132,212)(133,211)(134,210)(135,209)
(136,208)(137,207)(138,206)(139,205)(140,204)(141,203)(142,202)(143,201)
(144,200)(145,199)(146,198)(147,197)(148,196)(149,195)(150,194)(151,193)
(152,192)(153,191)(154,190)(155,189)(156,219)(157,218)(158,248)(159,247)
(160,246)(161,245)(162,244)(163,243)(164,242)(165,241)(166,240)(167,239)
(168,238)(169,237)(170,236)(171,235)(172,234)(173,233)(174,232)(175,231)
(176,230)(177,229)(178,228)(179,227)(180,226)(181,225)(182,224)(183,223)
(184,222)(185,221)(186,220);
poly := sub<Sym(248)|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,
s0*s1*s2*s1*s0*s1*s2*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*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*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