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G = C2×C24⋊D5  order 320 = 26·5

Direct product of C2 and C24⋊D5

direct product, non-abelian, soluble, monomial

Aliases: C2×C24⋊D5, C2≀D5, C25⋊D5, C24⋊D10, C24⋊C5⋊C22, (C2×C24⋊C5)⋊C2, SmallGroup(320,1636)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C24⋊C5 — C2×C24⋊D5
Chief series
C1 — C24 — C24⋊C5 — C24⋊D5 — C2×C24⋊D5
Lower central
C24⋊C5 — C2×C24⋊D5

Subgroups: 1434 in 193 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2 [×8], C4 [×6], C22 [×35], C5, C2×C4 [×12], D4 [×24], C23 [×33], D5 [×2], C10, C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×24], C24, C24 [×7], D10, C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C2×C22≀C2, C24⋊C5, C24⋊D5 [×2], C2×C24⋊C5, C2×C24⋊D5

Quotients:
C1, C2 [×3], C22, D5, D10, C24⋊D5, C2×C24⋊D5

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg=bc=cb, bd=db, fcf-1=gcg=be=eb, fbf-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf-1=bce, gdg=cde, fef-1=bcde, gfg=f-1 >

Permutation representations
►On 10 points - transitive group 10T23
Generators in S10
(1 8)(2 9)(3 10)(4 6)(5 7)

(1 8)(2 9)(4 6)(5 7)

(3 10)(4 6)

(3 10)(5 7)

(1 8)(3 10)(4 6)(5 7)

(1 2 3 4 5)(6 7 8 9 10)

(1 7)(2 6)(3 10)(4 9)(5 8)
 
G:=sub<Sym(10)| (1,8)(2,9)(3,10)(4,6)(5,7), (1,8)(2,9)(4,6)(5,7), (3,10)(4,6), (3,10)(5,7), (1,8)(3,10)(4,6)(5,7), (1,2,3,4,5)(6,7,8,9,10), (1,7)(2,6)(3,10)(4,9)(5,8)>;
 
G:=Group( (1,8)(2,9)(3,10)(4,6)(5,7), (1,8)(2,9)(4,6)(5,7), (3,10)(4,6), (3,10)(5,7), (1,8)(3,10)(4,6)(5,7), (1,2,3,4,5)(6,7,8,9,10), (1,7)(2,6)(3,10)(4,9)(5,8) );
 
G=PermutationGroup([(1,8),(2,9),(3,10),(4,6),(5,7)], [(1,8),(2,9),(4,6),(5,7)], [(3,10),(4,6)], [(3,10),(5,7)], [(1,8),(3,10),(4,6),(5,7)], [(1,2,3,4,5),(6,7,8,9,10)], [(1,7),(2,6),(3,10),(4,9),(5,8)])
 
G:=TransitiveGroup(10,23);
 

►On 20 points - transitive group 20T71
Generators in S20
(1 18)(2 19)(3 20)(4 16)(5 17)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 13)(2 14)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)

(3 15)(4 11)(6 16)(10 20)

(3 15)(5 12)(7 17)(10 20)

(1 13)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(10 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 12)(2 11)(3 15)(4 14)(5 13)(6 19)(7 18)(8 17)(9 16)(10 20)
 
G:=sub<Sym(20)| (1,18)(2,19)(3,20)(4,16)(5,17)(6,11)(7,12)(8,13)(9,14)(10,15), (1,13)(2,14)(4,11)(5,12)(6,16)(7,17)(8,18)(9,19), (3,15)(4,11)(6,16)(10,20), (3,15)(5,12)(7,17)(10,20), (1,13)(3,15)(4,11)(5,12)(6,16)(7,17)(8,18)(10,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,11)(3,15)(4,14)(5,13)(6,19)(7,18)(8,17)(9,16)(10,20)>;
 
G:=Group( (1,18)(2,19)(3,20)(4,16)(5,17)(6,11)(7,12)(8,13)(9,14)(10,15), (1,13)(2,14)(4,11)(5,12)(6,16)(7,17)(8,18)(9,19), (3,15)(4,11)(6,16)(10,20), (3,15)(5,12)(7,17)(10,20), (1,13)(3,15)(4,11)(5,12)(6,16)(7,17)(8,18)(10,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,11)(3,15)(4,14)(5,13)(6,19)(7,18)(8,17)(9,16)(10,20) );
 
G=PermutationGroup([(1,18),(2,19),(3,20),(4,16),(5,17),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,13),(2,14),(4,11),(5,12),(6,16),(7,17),(8,18),(9,19)], [(3,15),(4,11),(6,16),(10,20)], [(3,15),(5,12),(7,17),(10,20)], [(1,13),(3,15),(4,11),(5,12),(6,16),(7,17),(8,18),(10,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,19),(7,18),(8,17),(9,16),(10,20)])
 
G:=TransitiveGroup(20,71);
 

►On 20 points - transitive group 20T73
Generators in S20
(1 18)(2 19)(3 20)(4 16)(5 17)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 13)(5 12)(7 17)(8 18)

(2 14)(5 12)(7 17)(9 19)

(1 13)(2 14)(3 15)(5 12)(7 17)(8 18)(9 19)(10 20)

(4 11)(5 12)(6 16)(7 17)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)
 
G:=sub<Sym(20)| (1,18)(2,19)(3,20)(4,16)(5,17)(6,11)(7,12)(8,13)(9,14)(10,15), (1,13)(5,12)(7,17)(8,18), (2,14)(5,12)(7,17)(9,19), (1,13)(2,14)(3,15)(5,12)(7,17)(8,18)(9,19)(10,20), (4,11)(5,12)(6,16)(7,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)>;
 
G:=Group( (1,18)(2,19)(3,20)(4,16)(5,17)(6,11)(7,12)(8,13)(9,14)(10,15), (1,13)(5,12)(7,17)(8,18), (2,14)(5,12)(7,17)(9,19), (1,13)(2,14)(3,15)(5,12)(7,17)(8,18)(9,19)(10,20), (4,11)(5,12)(6,16)(7,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18) );
 
G=PermutationGroup([(1,18),(2,19),(3,20),(4,16),(5,17),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,13),(5,12),(7,17),(8,18)], [(2,14),(5,12),(7,17),(9,19)], [(1,13),(2,14),(3,15),(5,12),(7,17),(8,18),(9,19),(10,20)], [(4,11),(5,12),(6,16),(7,17)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,19),(17,18)])
 
G:=TransitiveGroup(20,73);
 

►On 20 points - transitive group 20T76
Generators in S20
(1 10)(2 6)(3 7)(4 8)(5 9)(11 16)(12 17)(13 18)(14 19)(15 20)

(1 15)(2 6)(4 8)(5 14)(9 19)(10 20)(11 16)(13 18)

(2 16)(3 7)(4 8)(5 19)(6 11)(9 14)(12 17)(13 18)

(1 20)(2 16)(3 12)(5 14)(6 11)(7 17)(9 19)(10 15)

(1 10)(3 7)(4 13)(5 14)(8 18)(9 19)(12 17)(15 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 19)(2 18)(3 17)(4 16)(5 20)(6 13)(7 12)(8 11)(9 15)(10 14)
 
G:=sub<Sym(20)| (1,10)(2,6)(3,7)(4,8)(5,9)(11,16)(12,17)(13,18)(14,19)(15,20), (1,15)(2,6)(4,8)(5,14)(9,19)(10,20)(11,16)(13,18), (2,16)(3,7)(4,8)(5,19)(6,11)(9,14)(12,17)(13,18), (1,20)(2,16)(3,12)(5,14)(6,11)(7,17)(9,19)(10,15), (1,10)(3,7)(4,13)(5,14)(8,18)(9,19)(12,17)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,19)(2,18)(3,17)(4,16)(5,20)(6,13)(7,12)(8,11)(9,15)(10,14)>;
 
G:=Group( (1,10)(2,6)(3,7)(4,8)(5,9)(11,16)(12,17)(13,18)(14,19)(15,20), (1,15)(2,6)(4,8)(5,14)(9,19)(10,20)(11,16)(13,18), (2,16)(3,7)(4,8)(5,19)(6,11)(9,14)(12,17)(13,18), (1,20)(2,16)(3,12)(5,14)(6,11)(7,17)(9,19)(10,15), (1,10)(3,7)(4,13)(5,14)(8,18)(9,19)(12,17)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,19)(2,18)(3,17)(4,16)(5,20)(6,13)(7,12)(8,11)(9,15)(10,14) );
 
G=PermutationGroup([(1,10),(2,6),(3,7),(4,8),(5,9),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,15),(2,6),(4,8),(5,14),(9,19),(10,20),(11,16),(13,18)], [(2,16),(3,7),(4,8),(5,19),(6,11),(9,14),(12,17),(13,18)], [(1,20),(2,16),(3,12),(5,14),(6,11),(7,17),(9,19),(10,15)], [(1,10),(3,7),(4,13),(5,14),(8,18),(9,19),(12,17),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,19),(2,18),(3,17),(4,16),(5,20),(6,13),(7,12),(8,11),(9,15),(10,14)])
 
G:=TransitiveGroup(20,76);
 

►On 20 points - transitive group 20T81
Generators in S20
(1 15)(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)(10 16)

(2 11)(3 12)(4 13)(5 14)(6 17)(7 18)(8 19)(9 20)

(1 15)(2 11)(6 17)(10 16)

(1 15)(3 12)(7 18)(10 16)

(1 15)(2 11)(3 12)(4 13)(6 17)(7 18)(8 19)(10 16)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 16)(2 20)(3 19)(4 18)(5 17)(6 14)(7 13)(8 12)(9 11)(10 15)
 
G:=sub<Sym(20)| (1,15)(2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20)(10,16), (2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20), (1,15)(2,11)(6,17)(10,16), (1,15)(3,12)(7,18)(10,16), (1,15)(2,11)(3,12)(4,13)(6,17)(7,18)(8,19)(10,16), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,14)(7,13)(8,12)(9,11)(10,15)>;
 
G:=Group( (1,15)(2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20)(10,16), (2,11)(3,12)(4,13)(5,14)(6,17)(7,18)(8,19)(9,20), (1,15)(2,11)(6,17)(10,16), (1,15)(3,12)(7,18)(10,16), (1,15)(2,11)(3,12)(4,13)(6,17)(7,18)(8,19)(10,16), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,14)(7,13)(8,12)(9,11)(10,15) );
 
G=PermutationGroup([(1,15),(2,11),(3,12),(4,13),(5,14),(6,17),(7,18),(8,19),(9,20),(10,16)], [(2,11),(3,12),(4,13),(5,14),(6,17),(7,18),(8,19),(9,20)], [(1,15),(2,11),(6,17),(10,16)], [(1,15),(3,12),(7,18),(10,16)], [(1,15),(2,11),(3,12),(4,13),(6,17),(7,18),(8,19),(10,16)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,14),(7,13),(8,12),(9,11),(10,15)])
 
G:=TransitiveGroup(20,81);
 

►On 20 points - transitive group 20T85
Generators in S20
(1 20)(2 16)(3 17)(4 18)(5 19)(6 11)(7 12)(8 13)(9 14)(10 15)

(1 15)(2 6)(4 8)(5 14)(9 19)(10 20)(11 16)(13 18)

(2 16)(3 7)(4 8)(5 19)(6 11)(9 14)(12 17)(13 18)

(1 20)(2 16)(3 12)(5 14)(6 11)(7 17)(9 19)(10 15)

(1 10)(3 7)(4 13)(5 14)(8 18)(9 19)(12 17)(15 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 8)(9 10)(11 13)(14 15)(16 18)(19 20)
 
G:=sub<Sym(20)| (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15), (1,15)(2,6)(4,8)(5,14)(9,19)(10,20)(11,16)(13,18), (2,16)(3,7)(4,8)(5,19)(6,11)(9,14)(12,17)(13,18), (1,20)(2,16)(3,12)(5,14)(6,11)(7,17)(9,19)(10,15), (1,10)(3,7)(4,13)(5,14)(8,18)(9,19)(12,17)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,8)(9,10)(11,13)(14,15)(16,18)(19,20)>;
 
G:=Group( (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15), (1,15)(2,6)(4,8)(5,14)(9,19)(10,20)(11,16)(13,18), (2,16)(3,7)(4,8)(5,19)(6,11)(9,14)(12,17)(13,18), (1,20)(2,16)(3,12)(5,14)(6,11)(7,17)(9,19)(10,15), (1,10)(3,7)(4,13)(5,14)(8,18)(9,19)(12,17)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,8)(9,10)(11,13)(14,15)(16,18)(19,20) );
 
G=PermutationGroup([(1,20),(2,16),(3,17),(4,18),(5,19),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,15),(2,6),(4,8),(5,14),(9,19),(10,20),(11,16),(13,18)], [(2,16),(3,7),(4,8),(5,19),(6,11),(9,14),(12,17),(13,18)], [(1,20),(2,16),(3,12),(5,14),(6,11),(7,17),(9,19),(10,15)], [(1,10),(3,7),(4,13),(5,14),(8,18),(9,19),(12,17),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,8),(9,10),(11,13),(14,15),(16,18),(19,20)])
 
G:=TransitiveGroup(20,85);
 

►On 20 points - transitive group 20T87
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 18)(7 19)(8 20)(9 16)(10 17)

(1 6)(2 19)(4 9)(5 17)(7 12)(10 15)(11 18)(14 16)

(1 11)(2 12)(3 20)(4 9)(5 15)(6 18)(7 19)(8 13)(10 17)(14 16)

(3 8)(4 14)(5 17)(9 16)(10 15)(13 20)

(1 18)(3 8)(4 16)(5 10)(6 11)(9 14)(13 20)(15 17)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 17)(7 16)(8 20)(9 19)(10 18)(11 15)(12 14)
 
G:=sub<Sym(20)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,18)(7,19)(8,20)(9,16)(10,17), (1,6)(2,19)(4,9)(5,17)(7,12)(10,15)(11,18)(14,16), (1,11)(2,12)(3,20)(4,9)(5,15)(6,18)(7,19)(8,13)(10,17)(14,16), (3,8)(4,14)(5,17)(9,16)(10,15)(13,20), (1,18)(3,8)(4,16)(5,10)(6,11)(9,14)(13,20)(15,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,17)(7,16)(8,20)(9,19)(10,18)(11,15)(12,14)>;
 
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,18)(7,19)(8,20)(9,16)(10,17), (1,6)(2,19)(4,9)(5,17)(7,12)(10,15)(11,18)(14,16), (1,11)(2,12)(3,20)(4,9)(5,15)(6,18)(7,19)(8,13)(10,17)(14,16), (3,8)(4,14)(5,17)(9,16)(10,15)(13,20), (1,18)(3,8)(4,16)(5,10)(6,11)(9,14)(13,20)(15,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,17)(7,16)(8,20)(9,19)(10,18)(11,15)(12,14) );
 
G=PermutationGroup([(1,11),(2,12),(3,13),(4,14),(5,15),(6,18),(7,19),(8,20),(9,16),(10,17)], [(1,6),(2,19),(4,9),(5,17),(7,12),(10,15),(11,18),(14,16)], [(1,11),(2,12),(3,20),(4,9),(5,15),(6,18),(7,19),(8,13),(10,17),(14,16)], [(3,8),(4,14),(5,17),(9,16),(10,15),(13,20)], [(1,18),(3,8),(4,16),(5,10),(6,11),(9,14),(13,20),(15,17)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,17),(7,16),(8,20),(9,19),(10,18),(11,15),(12,14)])
 
G:=TransitiveGroup(20,87);
 

Matrix representation ►G ⊆ GL5(ℤ)

-10000
0-1000
00-100
000-10
0000-1
,
-10000
01000
00100
00010
0000-1
,
10000
0-1000
00100
00010
0000-1
,
-10000
0-1000
00-100
00010
0000-1
,
10000
01000
00100
000-10
0000-1
,
00001
10000
01000
00100
00010
,
0000-1
000-10
00-100
0-1000
-10000

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,0] >;
 

Character table of C2×C24⋊D5

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F5A5B10A10B
 size 11555555202020202020202032323232
ρ111111111111111111111    trivial
ρ21-1-1111-1-11-1-111-1-1111-1-1    linear of order 2
ρ311111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ41-1-1111-1-1-111-1-111-111-1-1    linear of order 2
ρ52-2-2222-2-200000000-1+√5/2-1-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ62222222200000000-1-√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ72-2-2222-2-200000000-1-√5/2-1+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ82222222200000000-1+√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ95-5-111-33-1-111-11-1-110000    orthogonal faithful
ρ105-531-31-1-1-11-11-11-110000    orthogonal faithful
ρ11551-3111-311-1-1-1-1110000    orthogonal lifted from C24⋊D5
ρ12551-3111-3-1-11111-1-10000    orthogonal lifted from C24⋊D5
ρ135-531-31-1-11-11-11-11-10000    orthogonal faithful
ρ145-5-111-33-11-1-11-111-10000    orthogonal faithful
ρ1555111-3-31-1-1-1-111110000    orthogonal lifted from C24⋊D5
ρ1655-31-3111-1-111-1-1110000    orthogonal lifted from C24⋊D5
ρ175-5-1-311-131-11-1-11-110000    orthogonal faithful
ρ185-5-1-311-13-11-111-11-10000    orthogonal faithful
ρ1955-31-311111-1-111-1-10000    orthogonal lifted from C24⋊D5
ρ2055111-3-311111-1-1-1-10000    orthogonal lifted from C24⋊D5

In GAP, Magma, Sage, TeX 

C_2\times C_2^4\rtimes D_5
 
% in TeX
 
G:=Group("C2xC2^4:D5");
 
// GroupNames label
 
G:=SmallGroup(320,1636);
 
// by ID
 
G=gap.SmallGroup(320,1636);
 
# by ID
 
G:=PCGroup([7,-2,-2,-5,-2,2,2,2,338,1683,437,1068,9245,2539,4906,265]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^5=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,g*e*g=b*c=c*b,b*d=d*b,f*c*f^-1=g*c*g=b*e=e*b,f*b*f^-1=e,b*g=g*b,c*d=d*c,c*e=e*c,d*e=e*d,f*d*f^-1=b*c*e,g*d*g=c*d*e,f*e*f^-1=b*c*d*e,g*f*g=f^-1>;
 
// generators/relations
 

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