# Polytope of Type {2,220}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,220}*880
if this polytope has a name.
Group : SmallGroup(880,198)
Rank : 3
Schlafli Type : {2,220}
Number of vertices, edges, etc : 2, 220, 220
Order of s0s1s2 : 220
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,220,2} of size 1760
Vertex Figure Of :
{2,2,220} of size 1760
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,110}*440
4-fold quotients : {2,55}*220
5-fold quotients : {2,44}*176
10-fold quotients : {2,22}*88
11-fold quotients : {2,20}*80
20-fold quotients : {2,11}*44
22-fold quotients : {2,10}*40
44-fold quotients : {2,5}*20
55-fold quotients : {2,4}*16
110-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,220}*1760, {2,440}*1760
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  4, 13)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 14, 47)( 15, 57)( 16, 56)
( 17, 55)( 18, 54)( 19, 53)( 20, 52)( 21, 51)( 22, 50)( 23, 49)( 24, 48)
( 25, 36)( 26, 46)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 59, 68)( 60, 67)( 61, 66)( 62, 65)( 63, 64)
( 69,102)( 70,112)( 71,111)( 72,110)( 73,109)( 74,108)( 75,107)( 76,106)
( 77,105)( 78,104)( 79,103)( 80, 91)( 81,101)( 82,100)( 83, 99)( 84, 98)
( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)( 90, 92)(113,168)(114,178)
(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)
(123,169)(124,212)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)
(131,216)(132,215)(133,214)(134,213)(135,201)(136,211)(137,210)(138,209)
(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,190)
(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)
(155,192)(156,191)(157,179)(158,189)(159,188)(160,187)(161,186)(162,185)
(163,184)(164,183)(165,182)(166,181)(167,180);;
s2 := (  3,125)(  4,124)(  5,134)(  6,133)(  7,132)(  8,131)(  9,130)( 10,129)
( 11,128)( 12,127)( 13,126)( 14,114)( 15,113)( 16,123)( 17,122)( 18,121)
( 19,120)( 20,119)( 21,118)( 22,117)( 23,116)( 24,115)( 25,158)( 26,157)
( 27,167)( 28,166)( 29,165)( 30,164)( 31,163)( 32,162)( 33,161)( 34,160)
( 35,159)( 36,147)( 37,146)( 38,156)( 39,155)( 40,154)( 41,153)( 42,152)
( 43,151)( 44,150)( 45,149)( 46,148)( 47,136)( 48,135)( 49,145)( 50,144)
( 51,143)( 52,142)( 53,141)( 54,140)( 55,139)( 56,138)( 57,137)( 58,180)
( 59,179)( 60,189)( 61,188)( 62,187)( 63,186)( 64,185)( 65,184)( 66,183)
( 67,182)( 68,181)( 69,169)( 70,168)( 71,178)( 72,177)( 73,176)( 74,175)
( 75,174)( 76,173)( 77,172)( 78,171)( 79,170)( 80,213)( 81,212)( 82,222)
( 83,221)( 84,220)( 85,219)( 86,218)( 87,217)( 88,216)( 89,215)( 90,214)
( 91,202)( 92,201)( 93,211)( 94,210)( 95,209)( 96,208)( 97,207)( 98,206)
( 99,205)(100,204)(101,203)(102,191)(103,190)(104,200)(105,199)(106,198)
(107,197)(108,196)(109,195)(110,194)(111,193)(112,192);;
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*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(222)!(1,2);
s1 := Sym(222)!(  4, 13)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 14, 47)( 15, 57)
( 16, 56)( 17, 55)( 18, 54)( 19, 53)( 20, 52)( 21, 51)( 22, 50)( 23, 49)
( 24, 48)( 25, 36)( 26, 46)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 59, 68)( 60, 67)( 61, 66)( 62, 65)
( 63, 64)( 69,102)( 70,112)( 71,111)( 72,110)( 73,109)( 74,108)( 75,107)
( 76,106)( 77,105)( 78,104)( 79,103)( 80, 91)( 81,101)( 82,100)( 83, 99)
( 84, 98)( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)( 90, 92)(113,168)
(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)
(122,170)(123,169)(124,212)(125,222)(126,221)(127,220)(128,219)(129,218)
(130,217)(131,216)(132,215)(133,214)(134,213)(135,201)(136,211)(137,210)
(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)
(146,190)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)
(154,193)(155,192)(156,191)(157,179)(158,189)(159,188)(160,187)(161,186)
(162,185)(163,184)(164,183)(165,182)(166,181)(167,180);
s2 := Sym(222)!(  3,125)(  4,124)(  5,134)(  6,133)(  7,132)(  8,131)(  9,130)
( 10,129)( 11,128)( 12,127)( 13,126)( 14,114)( 15,113)( 16,123)( 17,122)
( 18,121)( 19,120)( 20,119)( 21,118)( 22,117)( 23,116)( 24,115)( 25,158)
( 26,157)( 27,167)( 28,166)( 29,165)( 30,164)( 31,163)( 32,162)( 33,161)
( 34,160)( 35,159)( 36,147)( 37,146)( 38,156)( 39,155)( 40,154)( 41,153)
( 42,152)( 43,151)( 44,150)( 45,149)( 46,148)( 47,136)( 48,135)( 49,145)
( 50,144)( 51,143)( 52,142)( 53,141)( 54,140)( 55,139)( 56,138)( 57,137)
( 58,180)( 59,179)( 60,189)( 61,188)( 62,187)( 63,186)( 64,185)( 65,184)
( 66,183)( 67,182)( 68,181)( 69,169)( 70,168)( 71,178)( 72,177)( 73,176)
( 74,175)( 75,174)( 76,173)( 77,172)( 78,171)( 79,170)( 80,213)( 81,212)
( 82,222)( 83,221)( 84,220)( 85,219)( 86,218)( 87,217)( 88,216)( 89,215)
( 90,214)( 91,202)( 92,201)( 93,211)( 94,210)( 95,209)( 96,208)( 97,207)
( 98,206)( 99,205)(100,204)(101,203)(102,191)(103,190)(104,200)(105,199)
(106,198)(107,197)(108,196)(109,195)(110,194)(111,193)(112,192);
poly := sub<Sym(222)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope