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## G = He5⋊C4order 500 = 22·53

### 2nd semidirect product of He5 and C4 acting faithfully

Aliases: He52C4, C52⋊Dic5, C522F5, C5⋊D5.D5, C52⋊C10.2C2, C5.2(D5.D5), SmallGroup(500,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — He5 — He5⋊C4
 Chief series C1 — C5 — C52 — He5 — C52⋊C10 — He5⋊C4
 Lower central He5 — He5⋊C4
 Upper central C1

Generators and relations for He5⋊C4
G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=a-1b-1c3, bc=cb, dbd-1=b3, dcd-1=c2 >

25C2
5C5
5C5
20C5
125C4
5D5
25C10
25D5
4C52
25F5
25F5
25Dic5

Character table of He5⋊C4

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B size 1 25 125 125 4 10 10 20 20 20 20 20 50 50 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ5 2 2 0 0 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 2 0 0 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 -2 0 0 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ8 2 -2 0 0 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ9 4 0 0 0 4 4 4 -1 -1 -1 -1 -1 0 0 orthogonal lifted from F5 ρ10 4 0 0 0 4 -1+√5 -1-√5 ζ53+2ζ5+1 2ζ54+ζ52+1 ζ54+2ζ53+1 -1 2ζ52+ζ5+1 0 0 complex lifted from D5.D5 ρ11 4 0 0 0 4 -1-√5 -1+√5 ζ54+2ζ53+1 2ζ52+ζ5+1 2ζ54+ζ52+1 -1 ζ53+2ζ5+1 0 0 complex lifted from D5.D5 ρ12 4 0 0 0 4 -1-√5 -1+√5 2ζ52+ζ5+1 ζ54+2ζ53+1 ζ53+2ζ5+1 -1 2ζ54+ζ52+1 0 0 complex lifted from D5.D5 ρ13 4 0 0 0 4 -1+√5 -1-√5 2ζ54+ζ52+1 ζ53+2ζ5+1 2ζ52+ζ5+1 -1 ζ54+2ζ53+1 0 0 complex lifted from D5.D5 ρ14 20 0 0 0 -5 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of He5⋊C4
On 25 points - transitive group 25T36
Generators in S25
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)
(1 13 21 18 9)(2 14 22 19 10)(3 15 23 20 6)(4 11 24 16 7)(5 12 25 17 8)
(2 10 19 22 14)(3 20 15 6 23)(4 24 7 11 16)(5 12 25 17 8)
(2 17 22 25)(3 4 15 24)(5 14)(6 16 23 7)(8 10 12 19)(9 18 13 21)(11 20)

G:=sub<Sym(25)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,13,21,18,9)(2,14,22,19,10)(3,15,23,20,6)(4,11,24,16,7)(5,12,25,17,8), (2,10,19,22,14)(3,20,15,6,23)(4,24,7,11,16)(5,12,25,17,8), (2,17,22,25)(3,4,15,24)(5,14)(6,16,23,7)(8,10,12,19)(9,18,13,21)(11,20)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,13,21,18,9)(2,14,22,19,10)(3,15,23,20,6)(4,11,24,16,7)(5,12,25,17,8), (2,10,19,22,14)(3,20,15,6,23)(4,24,7,11,16)(5,12,25,17,8), (2,17,22,25)(3,4,15,24)(5,14)(6,16,23,7)(8,10,12,19)(9,18,13,21)(11,20) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)], [(1,13,21,18,9),(2,14,22,19,10),(3,15,23,20,6),(4,11,24,16,7),(5,12,25,17,8)], [(2,10,19,22,14),(3,20,15,6,23),(4,24,7,11,16),(5,12,25,17,8)], [(2,17,22,25),(3,4,15,24),(5,14),(6,16,23,7),(8,10,12,19),(9,18,13,21),(11,20)]])

G:=TransitiveGroup(25,36);

On 25 points - transitive group 25T39
Generators in S25
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)
(1 25 9 19 12)(2 21 10 20 13)(3 22 6 16 14)(4 23 7 17 15)(5 24 8 18 11)
(1 2 14 7 11)(3 17 5 25 21)(4 8 19 20 6)(9 10 22 15 24)(12 13 16 23 18)
(2 7 11 14)(3 10 23 8)(4 5 22 13)(6 21 15 18)(9 12 19 25)(16 20 17 24)

G:=sub<Sym(25)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,25,9,19,12)(2,21,10,20,13)(3,22,6,16,14)(4,23,7,17,15)(5,24,8,18,11), (1,2,14,7,11)(3,17,5,25,21)(4,8,19,20,6)(9,10,22,15,24)(12,13,16,23,18), (2,7,11,14)(3,10,23,8)(4,5,22,13)(6,21,15,18)(9,12,19,25)(16,20,17,24)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,25,9,19,12)(2,21,10,20,13)(3,22,6,16,14)(4,23,7,17,15)(5,24,8,18,11), (1,2,14,7,11)(3,17,5,25,21)(4,8,19,20,6)(9,10,22,15,24)(12,13,16,23,18), (2,7,11,14)(3,10,23,8)(4,5,22,13)(6,21,15,18)(9,12,19,25)(16,20,17,24) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)], [(1,25,9,19,12),(2,21,10,20,13),(3,22,6,16,14),(4,23,7,17,15),(5,24,8,18,11)], [(1,2,14,7,11),(3,17,5,25,21),(4,8,19,20,6),(9,10,22,15,24),(12,13,16,23,18)], [(2,7,11,14),(3,10,23,8),(4,5,22,13),(6,21,15,18),(9,12,19,25),(16,20,17,24)]])

G:=TransitiveGroup(25,39);

Matrix representation of He5⋊C4 in GL20(ℤ)

 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
,
 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0

G:=sub<GL(20,Integers())| [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0],[0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0],[1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0] >;

He5⋊C4 in GAP, Magma, Sage, TeX

{\rm He}_5\rtimes C_4
% in TeX

G:=Group("He5:C4");
// GroupNames label

G:=SmallGroup(500,21);
// by ID

G=gap.SmallGroup(500,21);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1203,808,613,5004,5009]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a^-1*b^-1*c^3,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^2>;
// generators/relations

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