# Polytope of Type {14,6,4}

Atlas Canonical Name : {14,6,4}*672b
if this polytope has a name.
Group : SmallGroup(672,1260)
Rank : 4
Schlafli Type : {14,6,4}
Number of vertices, edges, etc : 14, 42, 12, 4
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{14,6,4,2} of size 1344
Vertex Figure Of :
{2,14,6,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
7-fold quotients : {2,6,4}*96c
14-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {14,12,4}*1344b, {14,12,4}*1344c, {28,6,4}*1344b, {14,6,4}*1344
Permutation Representation (GAP) :
```s0 := (  5, 25)(  6, 26)(  7, 27)(  8, 28)(  9, 21)( 10, 22)( 11, 23)( 12, 24)
( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 33, 53)( 34, 54)( 35, 55)( 36, 56)
( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 77)( 66, 78)( 67, 79)( 68, 80)
( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 89,109)( 90,110)( 91,111)( 92,112)
( 93,105)( 94,106)( 95,107)( 96,108)( 97,101)( 98,102)( 99,103)(100,104)
(117,137)(118,138)(119,139)(120,140)(121,133)(122,134)(123,135)(124,136)
(125,129)(126,130)(127,131)(128,132)(145,165)(146,166)(147,167)(148,168)
(149,161)(150,162)(151,163)(152,164)(153,157)(154,158)(155,159)(156,160);;
s1 := (  1,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 25)( 10, 27)( 11, 26)( 12, 28)
( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 29, 61)( 30, 63)( 31, 62)
( 32, 64)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 81)( 38, 83)( 39, 82)
( 40, 84)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 73)( 46, 75)( 47, 74)
( 48, 76)( 49, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)
( 56, 68)( 85, 89)( 86, 91)( 87, 90)( 88, 92)( 93,109)( 94,111)( 95,110)
( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(113,145)(114,147)
(115,146)(116,148)(117,141)(118,143)(119,142)(120,144)(121,165)(122,167)
(123,166)(124,168)(125,161)(126,163)(127,162)(128,164)(129,157)(130,159)
(131,158)(132,160)(133,153)(134,155)(135,154)(136,156)(137,149)(138,151)
(139,150)(140,152);;
s2 := (  1,113)(  2,114)(  3,116)(  4,115)(  5,117)(  6,118)(  7,120)(  8,119)
(  9,121)( 10,122)( 11,124)( 12,123)( 13,125)( 14,126)( 15,128)( 16,127)
( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)( 24,135)
( 25,137)( 26,138)( 27,140)( 28,139)( 29, 85)( 30, 86)( 31, 88)( 32, 87)
( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 93)( 38, 94)( 39, 96)( 40, 95)
( 41, 97)( 42, 98)( 43,100)( 44, 99)( 45,101)( 46,102)( 47,104)( 48,103)
( 49,105)( 50,106)( 51,108)( 52,107)( 53,109)( 54,110)( 55,112)( 56,111)
( 57,141)( 58,142)( 59,144)( 60,143)( 61,145)( 62,146)( 63,148)( 64,147)
( 65,149)( 66,150)( 67,152)( 68,151)( 69,153)( 70,154)( 71,156)( 72,155)
( 73,157)( 74,158)( 75,160)( 76,159)( 77,161)( 78,162)( 79,164)( 80,163)
( 81,165)( 82,166)( 83,168)( 84,167);;
s3 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*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(168)!(  5, 25)(  6, 26)(  7, 27)(  8, 28)(  9, 21)( 10, 22)( 11, 23)
( 12, 24)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 33, 53)( 34, 54)( 35, 55)
( 36, 56)( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 77)( 66, 78)( 67, 79)
( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 89,109)( 90,110)( 91,111)
( 92,112)( 93,105)( 94,106)( 95,107)( 96,108)( 97,101)( 98,102)( 99,103)
(100,104)(117,137)(118,138)(119,139)(120,140)(121,133)(122,134)(123,135)
(124,136)(125,129)(126,130)(127,131)(128,132)(145,165)(146,166)(147,167)
(148,168)(149,161)(150,162)(151,163)(152,164)(153,157)(154,158)(155,159)
(156,160);
s1 := Sym(168)!(  1,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 25)( 10, 27)( 11, 26)
( 12, 28)( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 29, 61)( 30, 63)
( 31, 62)( 32, 64)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 81)( 38, 83)
( 39, 82)( 40, 84)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 73)( 46, 75)
( 47, 74)( 48, 76)( 49, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 85, 89)( 86, 91)( 87, 90)( 88, 92)( 93,109)( 94,111)
( 95,110)( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(113,145)
(114,147)(115,146)(116,148)(117,141)(118,143)(119,142)(120,144)(121,165)
(122,167)(123,166)(124,168)(125,161)(126,163)(127,162)(128,164)(129,157)
(130,159)(131,158)(132,160)(133,153)(134,155)(135,154)(136,156)(137,149)
(138,151)(139,150)(140,152);
s2 := Sym(168)!(  1,113)(  2,114)(  3,116)(  4,115)(  5,117)(  6,118)(  7,120)
(  8,119)(  9,121)( 10,122)( 11,124)( 12,123)( 13,125)( 14,126)( 15,128)
( 16,127)( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)
( 24,135)( 25,137)( 26,138)( 27,140)( 28,139)( 29, 85)( 30, 86)( 31, 88)
( 32, 87)( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 93)( 38, 94)( 39, 96)
( 40, 95)( 41, 97)( 42, 98)( 43,100)( 44, 99)( 45,101)( 46,102)( 47,104)
( 48,103)( 49,105)( 50,106)( 51,108)( 52,107)( 53,109)( 54,110)( 55,112)
( 56,111)( 57,141)( 58,142)( 59,144)( 60,143)( 61,145)( 62,146)( 63,148)
( 64,147)( 65,149)( 66,150)( 67,152)( 68,151)( 69,153)( 70,154)( 71,156)
( 72,155)( 73,157)( 74,158)( 75,160)( 76,159)( 77,161)( 78,162)( 79,164)
( 80,163)( 81,165)( 82,166)( 83,168)( 84,167);
s3 := Sym(168)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167);
poly := sub<Sym(168)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*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