direct product, non-abelian, soluble, monomial, rational
Aliases: C3⋊S3×S4, C62⋊8D6, (C3×S4)⋊S3, C3⋊1(S3×S4), (C3×A4)⋊4D6, C32⋊7(C2×S4), (C32×S4)⋊2C2, C32⋊4S4⋊1C2, (C32×A4)⋊2C22, (C2×C6)⋊1S32, (A4×C3⋊S3)⋊1C2, A4⋊1(C2×C3⋊S3), C22⋊2(S3×C3⋊S3), (C22×C3⋊S3)⋊6S3, SmallGroup(432,746)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×A4 — C3⋊S3×S4 |
Generators and relations for C3⋊S3×S4
G = < a,b,c,d,e,f,g | a3=b3=c2=d2=e2=f3=g2=1, ab=ba, cac=a-1, ad=da, ae=ea, af=fa, ag=ga, cbc=b-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >
Subgroups: 2236 in 230 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, S4, S4, C2×A4, C22×S3, C33, C3⋊Dic3, C3×C12, S32, C3×A4, C3×A4, C2×C3⋊S3, C62, C62, S3×D4, C2×S4, S3×C32, C3×C3⋊S3, C33⋊C2, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C3×S4, C3⋊S4, S3×A4, C22×C3⋊S3, C22×C3⋊S3, S3×C3⋊S3, C32×A4, D4×C3⋊S3, S3×S4, C32×S4, C32⋊4S4, A4×C3⋊S3, C3⋊S3×S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, S32, C2×C3⋊S3, C2×S4, S3×C3⋊S3, S3×S4, C3⋊S3×S4
Character table of C3⋊S3×S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | |
| size | 1 | 3 | 6 | 9 | 27 | 54 | 2 | 2 | 2 | 2 | 8 | 16 | 16 | 16 | 16 | 6 | 54 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 72 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -2 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 0 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | 0 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | -2 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | 0 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ15 | 3 | -1 | -1 | -3 | 1 | 1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ17 | 3 | -1 | 1 | -3 | 1 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | -1 | 1 | 3 | -1 | 1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | -2 | 1 | 1 | -2 | 1 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ23 | 6 | -2 | -2 | 0 | 0 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3×S4 |
| ρ24 | 6 | -2 | -2 | 0 | 0 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3×S4 |
| ρ25 | 6 | -2 | 2 | 0 | 0 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 1 | 1 | 1 | -2 | -1 | 2 | -1 | -1 | 0 | -2 | 1 | 1 | 1 | orthogonal lifted from S3×S4 |
| ρ26 | 6 | -2 | -2 | 0 | 0 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3×S4 |
| ρ27 | 6 | -2 | 2 | 0 | 0 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 1 | -2 | 1 | 1 | -1 | -1 | -1 | 2 | 0 | 1 | 1 | -2 | 1 | orthogonal lifted from S3×S4 |
| ρ28 | 6 | -2 | -2 | 0 | 0 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from S3×S4 |
| ρ29 | 6 | -2 | 2 | 0 | 0 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 1 | 1 | -2 | 1 | 2 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | -2 | orthogonal lifted from S3×S4 |
| ρ30 | 6 | -2 | 2 | 0 | 0 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 1 | 1 | 1 | -1 | -1 | 2 | -1 | 0 | 1 | -2 | 1 | 1 | orthogonal lifted from S3×S4 |
►On 36 points
Generators in S36
(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 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 27 4)(2 25 5)(3 26 6)(7 17 29)(8 18 30)(9 16 28)(10 22 34)(11 23 35)(12 24 36)(13 21 33)(14 19 31)(15 20 32)
(2 3)(4 27)(5 26)(6 25)(7 28)(8 30)(9 29)(11 12)(13 14)(16 17)(19 33)(20 32)(21 31)(22 34)(23 36)(24 35)
(1 15)(2 13)(3 14)(4 32)(5 33)(6 31)(7 36)(8 34)(9 35)(10 18)(11 16)(12 17)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)
(1 18)(2 16)(3 17)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)(25 28)(26 29)(27 30)(31 36)(32 34)(33 35)
(7 36 31)(8 34 32)(9 35 33)(10 15 18)(11 13 16)(12 14 17)(19 29 24)(20 30 22)(21 28 23)
(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)(31 36)(32 34)(33 35)
G:=sub<Sym(36)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,27,4)(2,25,5)(3,26,6)(7,17,29)(8,18,30)(9,16,28)(10,22,34)(11,23,35)(12,24,36)(13,21,33)(14,19,31)(15,20,32), (2,3)(4,27)(5,26)(6,25)(7,28)(8,30)(9,29)(11,12)(13,14)(16,17)(19,33)(20,32)(21,31)(22,34)(23,36)(24,35), (1,15)(2,13)(3,14)(4,32)(5,33)(6,31)(7,36)(8,34)(9,35)(10,18)(11,16)(12,17)(19,26)(20,27)(21,25)(22,30)(23,28)(24,29), (1,18)(2,16)(3,17)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35), (7,36,31)(8,34,32)(9,35,33)(10,15,18)(11,13,16)(12,14,17)(19,29,24)(20,30,22)(21,28,23), (10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(31,36)(32,34)(33,35)>;
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)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,27,4)(2,25,5)(3,26,6)(7,17,29)(8,18,30)(9,16,28)(10,22,34)(11,23,35)(12,24,36)(13,21,33)(14,19,31)(15,20,32), (2,3)(4,27)(5,26)(6,25)(7,28)(8,30)(9,29)(11,12)(13,14)(16,17)(19,33)(20,32)(21,31)(22,34)(23,36)(24,35), (1,15)(2,13)(3,14)(4,32)(5,33)(6,31)(7,36)(8,34)(9,35)(10,18)(11,16)(12,17)(19,26)(20,27)(21,25)(22,30)(23,28)(24,29), (1,18)(2,16)(3,17)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35), (7,36,31)(8,34,32)(9,35,33)(10,15,18)(11,13,16)(12,14,17)(19,29,24)(20,30,22)(21,28,23), (10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(31,36)(32,34)(33,35) );
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),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,27,4),(2,25,5),(3,26,6),(7,17,29),(8,18,30),(9,16,28),(10,22,34),(11,23,35),(12,24,36),(13,21,33),(14,19,31),(15,20,32)], [(2,3),(4,27),(5,26),(6,25),(7,28),(8,30),(9,29),(11,12),(13,14),(16,17),(19,33),(20,32),(21,31),(22,34),(23,36),(24,35)], [(1,15),(2,13),(3,14),(4,32),(5,33),(6,31),(7,36),(8,34),(9,35),(10,18),(11,16),(12,17),(19,26),(20,27),(21,25),(22,30),(23,28),(24,29)], [(1,18),(2,16),(3,17),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(19,24),(20,22),(21,23),(25,28),(26,29),(27,30),(31,36),(32,34),(33,35)], [(7,36,31),(8,34,32),(9,35,33),(10,15,18),(11,13,16),(12,14,17),(19,29,24),(20,30,22),(21,28,23)], [(10,15),(11,13),(12,14),(19,24),(20,22),(21,23),(31,36),(32,34),(33,35)]])
Matrix representation of C3⋊S3×S4 ►in GL7(ℤ)
| -1 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(7,Integers())| [-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,-1,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
C3⋊S3×S4 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times S_4
% in TeX
G:=Group("C3:S3xS4"); // GroupNames label
G:=SmallGroup(432,746);
// by ID
G=gap.SmallGroup(432,746);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,93,675,2524,4548,2287,2659,3989]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^3=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
// generators/relations
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