direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×S3×D7, C21⋊C23, C14⋊1D6, C6⋊1D14, C42⋊C22, D42⋊5C2, D21⋊C22, (C6×D7)⋊3C2, (S3×C7)⋊C22, C7⋊1(C22×S3), (C3×D7)⋊C22, (S3×C14)⋊3C2, C3⋊1(C22×D7), SmallGroup(168,50)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C2×S3×D7 |
Generators and relations for C2×S3×D7
G = < a,b,c,d,e | a2=b3=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 336 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C7, C23, D6, D6, C2×C6, D7, D7, C14, C14, C21, C22×S3, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, C22×D7, S3×D7, C6×D7, S3×C14, D42, C2×S3×D7
Quotients: C1, C2, C22, S3, C23, D6, D7, C22×S3, D14, C22×D7, S3×D7, C2×S3×D7
Character table of C2×S3×D7
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 7A | 7B | 7C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 21A | 21B | 21C | 42A | 42B | 42C | |
| size | 1 | 1 | 3 | 3 | 7 | 7 | 21 | 21 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | -1 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | -1 | 1 | -1 | 1 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ23 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ25 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | orthogonal faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | orthogonal faithful |
| ρ28 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ29 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | orthogonal faithful |
►On 42 points
Generators in S42
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(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 37 38 39 40 41 42)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)
G:=sub<Sym(42)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (1,13,20)(2,14,21)(3,8,15)(4,9,16)(5,10,17)(6,11,18)(7,12,19)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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)(36,37,38,39,40,41,42), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)>;
G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (1,13,20)(2,14,21)(3,8,15)(4,9,16)(5,10,17)(6,11,18)(7,12,19)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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)(36,37,38,39,40,41,42), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41) );
G=PermutationGroup([[(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42)], [(1,13,20),(2,14,21),(3,8,15),(4,9,16),(5,10,17),(6,11,18),(7,12,19),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(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),(36,37,38,39,40,41,42)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41)]])
C2×S3×D7 is a maximal subgroup of
C28⋊D6 D6⋊D14
C2×S3×D7 is a maximal quotient of
D28⋊5S3 D28⋊S3 D12⋊D7 D84⋊C2 D21⋊Q8 D6.D14 D12⋊5D7 D14.D6 C28⋊D6 Dic7.D6 C42.C23 Dic3.D14 D6⋊D14
Matrix representation of C2×S3×D7 ►in GL5(𝔽43)
| 42 | 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 | 42 | 42 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 42 | 42 | 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 | 42 | 1 |
| 0 | 0 | 0 | 22 | 20 |
| 42 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 22 | 1 |
G:=sub<GL(5,GF(43))| [42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,42,1,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1],[42,0,0,0,0,0,1,42,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,42,22,0,0,0,1,20],[42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,42,22,0,0,0,0,1] >;
C2×S3×D7 in GAP, Magma, Sage, TeX
C_2\times S_3\times D_7
% in TeX
G:=Group("C2xS3xD7"); // GroupNames label
G:=SmallGroup(168,50);
// by ID
G=gap.SmallGroup(168,50);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,168,3604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
Export