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# Polytope of Type {252}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {252}*504
Also Known As : 252-gon, {252}. if this polytope has another name.
Group : SmallGroup(504,36)
Rank : 2
Schlafli Type : {252}
Number of vertices, edges, etc : 252, 252
Order of s0s1 : 252
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{252,2} of size 1008
Vertex Figure Of :
{2,252} of size 1008
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {126}*252
3-fold quotients : {84}*168
4-fold quotients : {63}*126
6-fold quotients : {42}*84
7-fold quotients : {36}*72
9-fold quotients : {28}*56
12-fold quotients : {21}*42
14-fold quotients : {18}*36
18-fold quotients : {14}*28
21-fold quotients : {12}*24
28-fold quotients : {9}*18
36-fold quotients : {7}*14
42-fold quotients : {6}*12
63-fold quotients : {4}*8
84-fold quotients : {3}*6
126-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {504}*1008
3-fold covers : {756}*1512
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)( 10, 13)
( 11, 15)( 12, 14)( 22, 44)( 23, 43)( 24, 45)( 25, 62)( 26, 61)( 27, 63)
( 28, 59)( 29, 58)( 30, 60)( 31, 56)( 32, 55)( 33, 57)( 34, 53)( 35, 52)
( 36, 54)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 65, 66)
( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)( 74, 78)
( 75, 77)( 85,107)( 86,106)( 87,108)( 88,125)( 89,124)( 90,126)( 91,122)
( 92,121)( 93,123)( 94,119)( 95,118)( 96,120)( 97,116)( 98,115)( 99,117)
(100,113)(101,112)(102,114)(103,110)(104,109)(105,111)(127,190)(128,192)
(129,191)(130,208)(131,210)(132,209)(133,205)(134,207)(135,206)(136,202)
(137,204)(138,203)(139,199)(140,201)(141,200)(142,196)(143,198)(144,197)
(145,193)(146,195)(147,194)(148,233)(149,232)(150,234)(151,251)(152,250)
(153,252)(154,248)(155,247)(156,249)(157,245)(158,244)(159,246)(160,242)
(161,241)(162,243)(163,239)(164,238)(165,240)(166,236)(167,235)(168,237)
(169,212)(170,211)(171,213)(172,230)(173,229)(174,231)(175,227)(176,226)
(177,228)(178,224)(179,223)(180,225)(181,221)(182,220)(183,222)(184,218)
(185,217)(186,219)(187,215)(188,214)(189,216);;
s1 := (  1,151)(  2,153)(  3,152)(  4,148)(  5,150)(  6,149)(  7,166)(  8,168)
(  9,167)( 10,163)( 11,165)( 12,164)( 13,160)( 14,162)( 15,161)( 16,157)
( 17,159)( 18,158)( 19,154)( 20,156)( 21,155)( 22,130)( 23,132)( 24,131)
( 25,127)( 26,129)( 27,128)( 28,145)( 29,147)( 30,146)( 31,142)( 32,144)
( 33,143)( 34,139)( 35,141)( 36,140)( 37,136)( 38,138)( 39,137)( 40,133)
( 41,135)( 42,134)( 43,173)( 44,172)( 45,174)( 46,170)( 47,169)( 48,171)
( 49,188)( 50,187)( 51,189)( 52,185)( 53,184)( 54,186)( 55,182)( 56,181)
( 57,183)( 58,179)( 59,178)( 60,180)( 61,176)( 62,175)( 63,177)( 64,214)
( 65,216)( 66,215)( 67,211)( 68,213)( 69,212)( 70,229)( 71,231)( 72,230)
( 73,226)( 74,228)( 75,227)( 76,223)( 77,225)( 78,224)( 79,220)( 80,222)
( 81,221)( 82,217)( 83,219)( 84,218)( 85,193)( 86,195)( 87,194)( 88,190)
( 89,192)( 90,191)( 91,208)( 92,210)( 93,209)( 94,205)( 95,207)( 96,206)
( 97,202)( 98,204)( 99,203)(100,199)(101,201)(102,200)(103,196)(104,198)
(105,197)(106,236)(107,235)(108,237)(109,233)(110,232)(111,234)(112,251)
(113,250)(114,252)(115,248)(116,247)(117,249)(118,245)(119,244)(120,246)
(121,242)(122,241)(123,243)(124,239)(125,238)(126,240);;
poly := Group([s0,s1]);;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1> := Group< s0,s1 | s0*s0, s1*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*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*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*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*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*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