Polytope of Type {12,42} with group SmallGroup(1008,903)

Overview

Group: SmallGroup(1008,903)
Rank: 3
Schläfli Type: {12,42}
Vertices, edges, …: 12, 252, 42
Order of s₀s₁s₂: 21
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat

Quotients maximal quotients in bold

3-fold

{4,42}*336c

6-fold

{4,21}*168

7-fold

{12,6}*144d

21-fold

{4,6}*48b

42-fold

{4,3}*24

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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