p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C23≀C2, C26⋊1C2, C24⋊15D4, C24⋊1C23, C25.88C22, C23.746C24, C22⋊3C22≀C2, (C22×C4)⋊1C23, C24⋊3C4⋊28C2, C23.628(C2×D4), (C22×D4)⋊15C22, C22.456(C22×D4), (C2×C22≀C2)⋊17C2, C2.29(C2×C22≀C2), (C2×C22⋊C4)⋊33C22, 2-Sylow(SO+(4,8)), SmallGroup(128,1578)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23≀C2
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, gag=ad=da, ae=ea, af=fa, bc=cb, bd=db, gbg=be=eb, bf=fb, cd=dc, ce=ec, gcg=cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 3332 in 1711 conjugacy classes, 180 normal (5 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C22⋊C4, C22×C4, C2×D4, C24, C24, C24, C2×C22⋊C4, C22≀C2, C22×D4, C25, C25, C24⋊3C4, C2×C22≀C2, C26, C23≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, C23≀C2
►On 16 points - transitive group 16T325
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 14)(2 13)(3 4)(5 6)(7 9)(8 10)(11 12)(15 16)
(1 7)(2 8)(3 5)(4 6)(9 14)(10 13)(11 15)(12 16)
(1 8)(2 7)(3 15)(4 16)(5 11)(6 12)(9 13)(10 14)
(1 9)(2 10)(3 11)(4 12)(5 15)(6 16)(7 14)(8 13)
(1 10)(2 9)(3 6)(4 5)(7 13)(8 14)(11 16)(12 15)
(1 3)(2 16)(4 7)(5 13)(6 10)(8 15)(9 11)(12 14)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,14)(2,13)(3,4)(5,6)(7,9)(8,10)(11,12)(15,16), (1,7)(2,8)(3,5)(4,6)(9,14)(10,13)(11,15)(12,16), (1,8)(2,7)(3,15)(4,16)(5,11)(6,12)(9,13)(10,14), (1,9)(2,10)(3,11)(4,12)(5,15)(6,16)(7,14)(8,13), (1,10)(2,9)(3,6)(4,5)(7,13)(8,14)(11,16)(12,15), (1,3)(2,16)(4,7)(5,13)(6,10)(8,15)(9,11)(12,14)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,14)(2,13)(3,4)(5,6)(7,9)(8,10)(11,12)(15,16), (1,7)(2,8)(3,5)(4,6)(9,14)(10,13)(11,15)(12,16), (1,8)(2,7)(3,15)(4,16)(5,11)(6,12)(9,13)(10,14), (1,9)(2,10)(3,11)(4,12)(5,15)(6,16)(7,14)(8,13), (1,10)(2,9)(3,6)(4,5)(7,13)(8,14)(11,16)(12,15), (1,3)(2,16)(4,7)(5,13)(6,10)(8,15)(9,11)(12,14) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,14),(2,13),(3,4),(5,6),(7,9),(8,10),(11,12),(15,16)], [(1,7),(2,8),(3,5),(4,6),(9,14),(10,13),(11,15),(12,16)], [(1,8),(2,7),(3,15),(4,16),(5,11),(6,12),(9,13),(10,14)], [(1,9),(2,10),(3,11),(4,12),(5,15),(6,16),(7,14),(8,13)], [(1,10),(2,9),(3,6),(4,5),(7,13),(8,14),(11,16),(12,15)], [(1,3),(2,16),(4,7),(5,13),(6,10),(8,15),(9,11),(12,14)]])
G:=TransitiveGroup(16,325);
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2AI | 2AJ | 4A | ··· | 4G |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 |
| kernel | C23≀C2 | C24⋊3C4 | C2×C22≀C2 | C26 | C24 |
| # reps | 1 | 7 | 7 | 1 | 28 |
Matrix representation of C23≀C2 ►in GL6(ℤ)
| -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 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 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 | 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 | -1 | 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 | -1 | 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 | 1 | 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 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | -2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,Integers())| [-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,-1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C23≀C2 in GAP, Magma, Sage, TeX
C_2^3\wr C_2
% in TeX
G:=Group("C2^3wrC2"); // GroupNames label
G:=SmallGroup(128,1578);
// by ID
G=gap.SmallGroup(128,1578);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,2019]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,a*b=b*a,a*c=c*a,g*a*g=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,g*b*g=b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,g*c*g=c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations