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# Polytope of Type {4,6,4}

Atlas Canonical Name : {4,6,4}*192a
Also Known As : {{4,6|2},{6,4|2}}. if this polytope has another name.
Group : SmallGroup(192,1149)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 4, 12, 12, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,4,2} of size 384
{4,6,4,4} of size 768
{4,6,4,6} of size 1152
{4,6,4,3} of size 1152
{4,6,4,6} of size 1728
{4,6,4,10} of size 1920
Vertex Figure Of :
{2,4,6,4} of size 384
{4,4,6,4} of size 768
{6,4,6,4} of size 1152
{3,4,6,4} of size 1152
{6,4,6,4} of size 1728
{10,4,6,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,4}*96a, {4,6,2}*96a
3-fold quotients : {4,2,4}*64
4-fold quotients : {2,6,2}*48
6-fold quotients : {2,2,4}*32, {4,2,2}*32
8-fold quotients : {2,3,2}*24
12-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,4}*384a, {4,6,8}*384a, {8,6,4}*384a
3-fold covers : {4,18,4}*576a, {4,6,12}*576a, {12,6,4}*576a, {4,6,12}*576c, {12,6,4}*576c
4-fold covers : {8,6,8}*768, {4,12,8}*768a, {8,12,4}*768a, {4,12,8}*768b, {8,12,4}*768b, {4,24,4}*768a, {4,12,4}*768a, {4,12,4}*768b, {4,24,4}*768b, {4,24,4}*768c, {4,24,4}*768d, {4,6,16}*768a, {16,6,4}*768a, {4,6,4}*768c, {4,6,4}*768d
5-fold covers : {4,6,20}*960a, {20,6,4}*960a, {4,30,4}*960a
6-fold covers : {4,36,4}*1152a, {4,12,12}*1152a, {4,12,12}*1152b, {12,12,4}*1152a, {12,12,4}*1152b, {4,18,8}*1152a, {8,18,4}*1152a, {8,6,12}*1152a, {12,6,8}*1152a, {8,6,12}*1152b, {12,6,8}*1152b, {4,6,24}*1152a, {24,6,4}*1152a, {4,6,24}*1152b, {24,6,4}*1152b
7-fold covers : {4,6,28}*1344a, {28,6,4}*1344a, {4,42,4}*1344a
9-fold covers : {4,54,4}*1728a, {4,6,36}*1728a, {36,6,4}*1728a, {4,18,12}*1728a, {12,18,4}*1728a, {4,6,12}*1728a, {12,6,4}*1728a, {4,18,12}*1728b, {12,18,4}*1728b, {4,6,12}*1728c, {12,6,4}*1728c, {12,6,12}*1728b, {12,6,12}*1728c, {12,6,12}*1728d, {12,6,12}*1728g, {4,6,12}*1728h, {12,6,4}*1728h, {4,6,4}*1728c, {4,6,4}*1728d
10-fold covers : {4,60,4}*1920a, {4,12,20}*1920a, {20,12,4}*1920a, {4,30,8}*1920a, {8,30,4}*1920a, {8,6,20}*1920, {20,6,8}*1920, {4,6,40}*1920a, {40,6,4}*1920a
Permutation Representation (GAP) :
```s0 := ( 1,49)( 2,50)( 3,51)( 4,52)( 5,53)( 6,54)( 7,55)( 8,56)( 9,57)(10,58)
(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)
(22,70)(23,71)(24,72)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)
(33,75)(34,76)(35,77)(36,78)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,85)
(44,86)(45,87)(46,88)(47,89)(48,90);;
s1 := ( 1,25)( 2,27)( 3,26)( 4,28)( 5,30)( 6,29)( 7,31)( 8,33)( 9,32)(10,34)
(11,36)(12,35)(13,43)(14,45)(15,44)(16,46)(17,48)(18,47)(19,37)(20,39)(21,38)
(22,40)(23,42)(24,41)(49,73)(50,75)(51,74)(52,76)(53,78)(54,77)(55,79)(56,81)
(57,80)(58,82)(59,84)(60,83)(61,91)(62,93)(63,92)(64,94)(65,96)(66,95)(67,85)
(68,87)(69,86)(70,88)(71,90)(72,89);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,23)(14,22)(15,24)(16,20)(17,19)(18,21)
(25,26)(28,29)(31,32)(34,35)(37,47)(38,46)(39,48)(40,44)(41,43)(42,45)(49,50)
(52,53)(55,56)(58,59)(61,71)(62,70)(63,72)(64,68)(65,67)(66,69)(73,74)(76,77)
(79,80)(82,83)(85,95)(86,94)(87,96)(88,92)(89,91)(90,93);;
s3 := ( 1,61)( 2,62)( 3,63)( 4,64)( 5,65)( 6,66)( 7,67)( 8,68)( 9,69)(10,70)
(11,71)(12,72)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)
(22,58)(23,59)(24,60)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)(32,86)
(33,87)(34,88)(35,89)(36,90)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)
(44,74)(45,75)(46,76)(47,77)(48,78);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(96)!( 1,49)( 2,50)( 3,51)( 4,52)( 5,53)( 6,54)( 7,55)( 8,56)( 9,57)
(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)
(21,69)(22,70)(23,71)(24,72)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)
(32,74)(33,75)(34,76)(35,77)(36,78)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)
(43,85)(44,86)(45,87)(46,88)(47,89)(48,90);
s1 := Sym(96)!( 1,25)( 2,27)( 3,26)( 4,28)( 5,30)( 6,29)( 7,31)( 8,33)( 9,32)
(10,34)(11,36)(12,35)(13,43)(14,45)(15,44)(16,46)(17,48)(18,47)(19,37)(20,39)
(21,38)(22,40)(23,42)(24,41)(49,73)(50,75)(51,74)(52,76)(53,78)(54,77)(55,79)
(56,81)(57,80)(58,82)(59,84)(60,83)(61,91)(62,93)(63,92)(64,94)(65,96)(66,95)
(67,85)(68,87)(69,86)(70,88)(71,90)(72,89);
s2 := Sym(96)!( 1, 2)( 4, 5)( 7, 8)(10,11)(13,23)(14,22)(15,24)(16,20)(17,19)
(18,21)(25,26)(28,29)(31,32)(34,35)(37,47)(38,46)(39,48)(40,44)(41,43)(42,45)
(49,50)(52,53)(55,56)(58,59)(61,71)(62,70)(63,72)(64,68)(65,67)(66,69)(73,74)
(76,77)(79,80)(82,83)(85,95)(86,94)(87,96)(88,92)(89,91)(90,93);
s3 := Sym(96)!( 1,61)( 2,62)( 3,63)( 4,64)( 5,65)( 6,66)( 7,67)( 8,68)( 9,69)
(10,70)(11,71)(12,72)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)
(21,57)(22,58)(23,59)(24,60)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,85)
(32,86)(33,87)(34,88)(35,89)(36,90)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)
(43,73)(44,74)(45,75)(46,76)(47,77)(48,78);
poly := sub<Sym(96)|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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope