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# Polytope of Type {4,21}

Atlas Canonical Name : {4,21}*336
if this polytope has a name.
Group : SmallGroup(336,215)
Rank : 3
Schlafli Type : {4,21}
Number of vertices, edges, etc : 8, 84, 42
Order of s0s1s2 : 42
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,21,2} of size 672
{4,21,4} of size 1344
Vertex Figure Of :
{2,4,21} of size 672
{4,4,21} of size 1344
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,21}*168
4-fold quotients : {2,21}*84
7-fold quotients : {4,3}*48
12-fold quotients : {2,7}*28
14-fold quotients : {4,3}*24
28-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,21}*672, {4,42}*672
3-fold covers : {4,63}*1008, {12,21}*1008
4-fold covers : {8,21}*1344, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
5-fold covers : {4,105}*1680
Permutation Representation (GAP) :
```s0 := (  1, 87)(  2, 88)(  3, 85)(  4, 86)(  5, 91)(  6, 92)(  7, 89)(  8, 90)
(  9, 95)( 10, 96)( 11, 93)( 12, 94)( 13, 99)( 14,100)( 15, 97)( 16, 98)
( 17,103)( 18,104)( 19,101)( 20,102)( 21,107)( 22,108)( 23,105)( 24,106)
( 25,111)( 26,112)( 27,109)( 28,110)( 29,115)( 30,116)( 31,113)( 32,114)
( 33,119)( 34,120)( 35,117)( 36,118)( 37,123)( 38,124)( 39,121)( 40,122)
( 41,127)( 42,128)( 43,125)( 44,126)( 45,131)( 46,132)( 47,129)( 48,130)
( 49,135)( 50,136)( 51,133)( 52,134)( 53,139)( 54,140)( 55,137)( 56,138)
( 57,143)( 58,144)( 59,141)( 60,142)( 61,147)( 62,148)( 63,145)( 64,146)
( 65,151)( 66,152)( 67,149)( 68,150)( 69,155)( 70,156)( 71,153)( 72,154)
( 73,159)( 74,160)( 75,157)( 76,158)( 77,163)( 78,164)( 79,161)( 80,162)
( 81,167)( 82,168)( 83,165)( 84,166);;
s1 := (  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)( 11, 22)
( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)( 31, 58)
( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)( 39, 78)
( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)
( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)( 94,107)
( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)(114,143)
(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)(122,163)
(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)(130,155)
(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)(138,147)
(139,146)(140,148);;
s2 := (  1, 61)(  2, 64)(  3, 63)(  4, 62)(  5, 57)(  6, 60)(  7, 59)(  8, 58)
(  9, 81)( 10, 84)( 11, 83)( 12, 82)( 13, 77)( 14, 80)( 15, 79)( 16, 78)
( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 69)( 22, 72)( 23, 71)( 24, 70)
( 25, 65)( 26, 68)( 27, 67)( 28, 66)( 29, 33)( 30, 36)( 31, 35)( 32, 34)
( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)( 43, 51)( 44, 50)
( 46, 48)( 85,145)( 86,148)( 87,147)( 88,146)( 89,141)( 90,144)( 91,143)
( 92,142)( 93,165)( 94,168)( 95,167)( 96,166)( 97,161)( 98,164)( 99,163)
(100,162)(101,157)(102,160)(103,159)(104,158)(105,153)(106,156)(107,155)
(108,154)(109,149)(110,152)(111,151)(112,150)(113,117)(114,120)(115,119)
(116,118)(121,137)(122,140)(123,139)(124,138)(125,133)(126,136)(127,135)
(128,134)(130,132);;
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*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope