direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C2≀C4, C25⋊2C4, C24.34D4, C24⋊5(C2×C4), C22≀C2⋊5C4, C23.1(C2×D4), (C2×D4).123D4, C23⋊C4⋊2C22, (C22×C4).88D4, (C2×D4).12C23, C4.D4⋊17C22, C23.51(C22×C4), C22≀C2.19C22, C22.48(C23⋊C4), (C22×D4).98C22, C23.201(C22⋊C4), (C2×C4).1(C2×D4), C22⋊C4⋊1(C2×C4), (C2×C22⋊C4)⋊7C4, (C2×C23⋊C4)⋊11C2, C2.30(C2×C23⋊C4), (C2×C22≀C2).3C2, (C2×D4).121(C2×C4), (C2×C4.D4)⋊26C2, (C2×C4).23(C22⋊C4), C22.54(C2×C22⋊C4), SmallGroup(128,850)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C2≀C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >
Subgroups: 660 in 219 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C23⋊C4, C23⋊C4, C4.D4, C4.D4, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22≀C2, C2×M4(2), C22×D4, C22×D4, C25, C2≀C4, C2×C23⋊C4, C2×C4.D4, C2×C22≀C2, C2×C2≀C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2≀C4, C2×C23⋊C4, C2×C2≀C4
Character table of C2×C2≀C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | i | -i | 1 | i | 1 | -i | i | -i | -i | i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -i | -i | -1 | i | 1 | i | i | -i | i | -i | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | i | i | -1 | -i | 1 | -i | -i | i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -i | i | 1 | -i | 1 | i | -i | i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | -i | 1 | i | -1 | i | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | -1 | i | -1 | -i | -i | i | i | -i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | -1 | -i | -1 | i | i | -i | -i | i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | i | 1 | -i | -1 | -i | i | -i | i | -i | linear of order 4 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ23 | 4 | -4 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ24 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
►On 16 points - transitive group 16T227
Generators in S16
(1 15)(2 16)(3 13)(4 14)(5 7)(6 8)(9 11)(10 12)
(1 3)(2 7)(4 11)(5 16)(6 10)(8 12)(9 14)(13 15)
(1 8)(2 11)(3 12)(4 7)(5 14)(6 15)(9 16)(10 13)
(2 14)(4 16)(5 11)(7 9)
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,15)(2,16)(3,13)(4,14)(5,7)(6,8)(9,11)(10,12), (1,3)(2,7)(4,11)(5,16)(6,10)(8,12)(9,14)(13,15), (1,8)(2,11)(3,12)(4,7)(5,14)(6,15)(9,16)(10,13), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,7)(6,8)(9,11)(10,12), (1,3)(2,7)(4,11)(5,16)(6,10)(8,12)(9,14)(13,15), (1,8)(2,11)(3,12)(4,7)(5,14)(6,15)(9,16)(10,13), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,7),(6,8),(9,11),(10,12)], [(1,3),(2,7),(4,11),(5,16),(6,10),(8,12),(9,14),(13,15)], [(1,8),(2,11),(3,12),(4,7),(5,14),(6,15),(9,16),(10,13)], [(2,14),(4,16),(5,11),(7,9)], [(1,13),(2,14),(3,15),(4,16),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,227);
►On 16 points - transitive group 16T259
Generators in S16
(1 10)(2 11)(3 12)(4 9)(5 16)(6 13)(7 14)(8 15)
(1 10)(4 7)(8 15)(9 14)
(1 15)(2 11)(4 7)(5 16)(8 10)(9 14)
(1 10)(2 5)(3 12)(4 7)(6 13)(8 15)(9 14)(11 16)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,9)(5,16)(6,13)(7,14)(8,15), (1,10)(4,7)(8,15)(9,14), (1,15)(2,11)(4,7)(5,16)(8,10)(9,14), (1,10)(2,5)(3,12)(4,7)(6,13)(8,15)(9,14)(11,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,16)(6,13)(7,14)(8,15), (1,10)(4,7)(8,15)(9,14), (1,15)(2,11)(4,7)(5,16)(8,10)(9,14), (1,10)(2,5)(3,12)(4,7)(6,13)(8,15)(9,14)(11,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,16),(6,13),(7,14),(8,15)], [(1,10),(4,7),(8,15),(9,14)], [(1,15),(2,11),(4,7),(5,16),(8,10),(9,14)], [(1,10),(2,5),(3,12),(4,7),(6,13),(8,15),(9,14),(11,16)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,259);
►On 16 points - transitive group 16T261
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(2 15)(7 12)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (2,15)(7,12), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (2,15)(7,12), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(2,15),(7,12)], [(2,15),(3,16),(7,12),(8,9)], [(2,15),(4,13),(5,10),(7,12)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,261);
►On 16 points - transitive group 16T273
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)
(1 5)(2 16)(3 7)(6 12)(9 13)(11 15)
(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14), (1,5)(2,16)(3,7)(6,12)(9,13)(11,15), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14), (1,5)(2,16)(3,7)(6,12)(9,13)(11,15), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,15),(6,16),(7,13),(8,14)], [(1,5),(2,16),(3,7),(6,12),(9,13),(11,15)], [(1,5),(2,12),(3,9),(4,8),(6,16),(7,13),(10,14),(11,15)], [(2,16),(4,14),(6,12),(8,10)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,273);
►On 16 points - transitive group 16T283
Generators in S16
(1 6)(2 5)(3 8)(4 7)(9 14)(10 15)(11 16)(12 13)
(1 11)(2 8)(3 5)(4 14)(6 16)(7 9)(10 12)(13 15)
(1 9)(2 10)(3 13)(4 16)(5 15)(6 14)(7 11)(8 12)
(1 7)(4 6)(9 11)(14 16)
(1 7)(2 8)(3 5)(4 6)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,6)(2,5)(3,8)(4,7)(9,14)(10,15)(11,16)(12,13), (1,11)(2,8)(3,5)(4,14)(6,16)(7,9)(10,12)(13,15), (1,9)(2,10)(3,13)(4,16)(5,15)(6,14)(7,11)(8,12), (1,7)(4,6)(9,11)(14,16), (1,7)(2,8)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,6)(2,5)(3,8)(4,7)(9,14)(10,15)(11,16)(12,13), (1,11)(2,8)(3,5)(4,14)(6,16)(7,9)(10,12)(13,15), (1,9)(2,10)(3,13)(4,16)(5,15)(6,14)(7,11)(8,12), (1,7)(4,6)(9,11)(14,16), (1,7)(2,8)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,6),(2,5),(3,8),(4,7),(9,14),(10,15),(11,16),(12,13)], [(1,11),(2,8),(3,5),(4,14),(6,16),(7,9),(10,12),(13,15)], [(1,9),(2,10),(3,13),(4,16),(5,15),(6,14),(7,11),(8,12)], [(1,7),(4,6),(9,11),(14,16)], [(1,7),(2,8),(3,5),(4,6),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,283);
Matrix representation of C2×C2≀C4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,16,0,0] >;
C2×C2≀C4 in GAP, Magma, Sage, TeX
C_2\times C_2\wr C_4
% in TeX
G:=Group("C2xC2wrC4"); // GroupNames label
G:=SmallGroup(128,850);
// by ID
G=gap.SmallGroup(128,850);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,851,375,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations
Export