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

Overview

Group: SmallGroup(1008,903)
Rank: 3
Schläfli Type: {42,12}
Vertices, edges, …: 42, 252, 12
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

{42,4}*336c

6-fold

{21,4}*168

7-fold

{6,12}*144d

21-fold

{6,4}*48b

42-fold

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

Permutation Representation (Magma)

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

References