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# Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*288a
Also Known As : {12,12|2}. if this polytope has another name.
Group : SmallGroup(288,571)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 12, 72, 12
Order of s0s1s2 : 12
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,12,2} of size 576
{12,12,4} of size 1152
{12,12,3} of size 1152
{12,12,4} of size 1152
{12,12,4} of size 1152
{12,12,6} of size 1728
{12,12,6} of size 1728
{12,12,6} of size 1728
{12,12,3} of size 1728
Vertex Figure Of :
{2,12,12} of size 576
{4,12,12} of size 1152
{3,12,12} of size 1152
{4,12,12} of size 1152
{4,12,12} of size 1152
{6,12,12} of size 1728
{6,12,12} of size 1728
{6,12,12} of size 1728
{3,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*144a, {12,6}*144a
3-fold quotients : {4,12}*96a, {12,4}*96a
4-fold quotients : {6,6}*72a
6-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
9-fold quotients : {4,4}*32
12-fold quotients : {2,6}*24, {6,2}*24
18-fold quotients : {2,4}*16, {4,2}*16
24-fold quotients : {2,3}*12, {3,2}*12
36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*576a, {12,24}*576c, {24,12}*576c, {12,24}*576e, {24,12}*576e
3-fold covers : {12,36}*864a, {36,12}*864a, {12,12}*864c, {12,12}*864h
4-fold covers : {12,24}*1152b, {24,12}*1152b, {24,24}*1152b, {24,24}*1152g, {24,24}*1152i, {24,24}*1152k, {12,48}*1152b, {48,12}*1152b, {12,48}*1152e, {48,12}*1152e, {12,24}*1152e, {24,12}*1152e, {12,12}*1152c, {12,12}*1152k, {12,12}*1152n
5-fold covers : {12,60}*1440b, {60,12}*1440b
6-fold covers : {12,36}*1728a, {36,12}*1728a, {12,12}*1728c, {12,72}*1728a, {72,12}*1728a, {24,36}*1728c, {36,24}*1728c, {12,24}*1728d, {24,12}*1728d, {12,72}*1728c, {72,12}*1728c, {24,36}*1728d, {36,24}*1728d, {12,24}*1728f, {24,12}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)
(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)(10,49)
(11,51)(12,50)(13,46)(14,48)(15,47)(16,52)(17,54)(18,53)(19,58)(20,60)(21,59)
(22,55)(23,57)(24,56)(25,61)(26,63)(27,62)(28,67)(29,69)(30,68)(31,64)(32,66)
(33,65)(34,70)(35,72)(36,71);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)
(31,32)(34,35)(37,47)(38,46)(39,48)(40,50)(41,49)(42,51)(43,53)(44,52)(45,54)
(55,65)(56,64)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72);;
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*s2*s1*s0*s1*s2*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,
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(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)
(45,60)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69);
s1 := Sym(72)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)
(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)(16,52)(17,54)(18,53)(19,58)(20,60)
(21,59)(22,55)(23,57)(24,56)(25,61)(26,63)(27,62)(28,67)(29,69)(30,68)(31,64)
(32,66)(33,65)(34,70)(35,72)(36,71);
s2 := Sym(72)!( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)
(28,29)(31,32)(34,35)(37,47)(38,46)(39,48)(40,50)(41,49)(42,51)(43,53)(44,52)
(45,54)(55,65)(56,64)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72);
poly := sub<Sym(72)|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*s2*s1*s0*s1*s2*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,
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