# Polytope of Type {14,4,12}

Atlas Canonical Name : {14,4,12}*1344
Also Known As : {{14,4|2},{4,12|2}}. if this polytope has another name.
Group : SmallGroup(1344,7764)
Rank : 4
Schlafli Type : {14,4,12}
Number of vertices, edges, etc : 14, 28, 24, 12
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {14,2,12}*672, {14,4,6}*672
3-fold quotients : {14,4,4}*448
4-fold quotients : {7,2,12}*336, {14,2,6}*336
6-fold quotients : {14,2,4}*224, {14,4,2}*224
7-fold quotients : {2,4,12}*192a
8-fold quotients : {7,2,6}*168, {14,2,3}*168
12-fold quotients : {7,2,4}*112, {14,2,2}*112
14-fold quotients : {2,2,12}*96, {2,4,6}*96a
16-fold quotients : {7,2,3}*84
21-fold quotients : {2,4,4}*64
24-fold quotients : {7,2,2}*56
28-fold quotients : {2,2,6}*48
42-fold quotients : {2,2,4}*32, {2,4,2}*32
56-fold quotients : {2,2,3}*24
84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)
(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166);;
s1 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,128)( 86,127)( 87,133)( 88,132)
( 89,131)( 90,130)( 91,129)( 92,135)( 93,134)( 94,140)( 95,139)( 96,138)
( 97,137)( 98,136)( 99,142)(100,141)(101,147)(102,146)(103,145)(104,144)
(105,143)(106,149)(107,148)(108,154)(109,153)(110,152)(111,151)(112,150)
(113,156)(114,155)(115,161)(116,160)(117,159)(118,158)(119,157)(120,163)
(121,162)(122,168)(123,167)(124,166)(125,165)(126,164);;
s2 := (  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)(  8, 99)
(  9,100)( 10,101)( 11,102)( 12,103)( 13,104)( 14,105)( 15, 92)( 16, 93)
( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22,106)( 23,107)( 24,108)
( 25,109)( 26,110)( 27,111)( 28,112)( 29,120)( 30,121)( 31,122)( 32,123)
( 33,124)( 34,125)( 35,126)( 36,113)( 37,114)( 38,115)( 39,116)( 40,117)
( 41,118)( 42,119)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)( 48,132)
( 49,133)( 50,141)( 51,142)( 52,143)( 53,144)( 54,145)( 55,146)( 56,147)
( 57,134)( 58,135)( 59,136)( 60,137)( 61,138)( 62,139)( 63,140)( 64,148)
( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,162)( 72,163)
( 73,164)( 74,165)( 75,166)( 76,167)( 77,168)( 78,155)( 79,156)( 80,157)
( 81,158)( 82,159)( 83,160)( 84,161);;
s3 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 22, 29)
( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85,113)( 86,114)( 87,115)( 88,116)
( 89,117)( 90,118)( 91,119)( 92,106)( 93,107)( 94,108)( 95,109)( 96,110)
( 97,111)( 98,112)( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)
(105,126)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)
(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)(141,162)
(142,163)(143,164)(144,165)(145,166)(146,167)(147,168);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*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)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)
(129,132)(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)
(149,154)(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)
(165,166);
s1 := Sym(168)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)
( 53, 55)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,128)( 86,127)( 87,133)
( 88,132)( 89,131)( 90,130)( 91,129)( 92,135)( 93,134)( 94,140)( 95,139)
( 96,138)( 97,137)( 98,136)( 99,142)(100,141)(101,147)(102,146)(103,145)
(104,144)(105,143)(106,149)(107,148)(108,154)(109,153)(110,152)(111,151)
(112,150)(113,156)(114,155)(115,161)(116,160)(117,159)(118,158)(119,157)
(120,163)(121,162)(122,168)(123,167)(124,166)(125,165)(126,164);
s2 := Sym(168)!(  1, 85)(  2, 86)(  3, 87)(  4, 88)(  5, 89)(  6, 90)(  7, 91)
(  8, 99)(  9,100)( 10,101)( 11,102)( 12,103)( 13,104)( 14,105)( 15, 92)
( 16, 93)( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22,106)( 23,107)
( 24,108)( 25,109)( 26,110)( 27,111)( 28,112)( 29,120)( 30,121)( 31,122)
( 32,123)( 33,124)( 34,125)( 35,126)( 36,113)( 37,114)( 38,115)( 39,116)
( 40,117)( 41,118)( 42,119)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)
( 48,132)( 49,133)( 50,141)( 51,142)( 52,143)( 53,144)( 54,145)( 55,146)
( 56,147)( 57,134)( 58,135)( 59,136)( 60,137)( 61,138)( 62,139)( 63,140)
( 64,148)( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,162)
( 72,163)( 73,164)( 74,165)( 75,166)( 76,167)( 77,168)( 78,155)( 79,156)
( 80,157)( 81,158)( 82,159)( 83,160)( 84,161);
s3 := Sym(168)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 22, 29)( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85,113)( 86,114)( 87,115)
( 88,116)( 89,117)( 90,118)( 91,119)( 92,106)( 93,107)( 94,108)( 95,109)
( 96,110)( 97,111)( 98,112)( 99,120)(100,121)(101,122)(102,123)(103,124)
(104,125)(105,126)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)
(133,161)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)
(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*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