direct product, p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C2×C2≀C22, C24⋊4D4, C24⋊6C23, C25⋊3C22, C23.5C24, 2+ (1+4)⋊7C22, C23⋊(C2×D4), (C2×D4)⋊24D4, (C22×C4)⋊5D4, C23⋊C4⋊5C22, C22⋊C4⋊1C23, (C2×D4).39C23, C22≀C2⋊23C22, C22.25C22≀C2, (C2×2+ (1+4))⋊5C2, C22.39(C22×D4), (C22×D4).332C22, (C2×C4)⋊(C2×D4), (C2×C23⋊C4)⋊16C2, (C2×C22≀C2)⋊19C2, C2.60(C2×C22≀C2), (C2×C22⋊C4)⋊36C22, SmallGroup(128,1755)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 1236 in 509 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×16], C4 [×12], C22, C22 [×6], C22 [×71], C2×C4 [×6], C2×C4 [×30], D4 [×60], Q8 [×4], C23, C23 [×12], C23 [×64], C22⋊C4 [×6], C22⋊C4 [×15], C22×C4 [×3], C22×C4 [×6], C2×D4 [×12], C2×D4 [×63], C2×Q8, C4○D4 [×24], C24, C24 [×5], C24 [×11], C23⋊C4 [×12], C2×C22⋊C4 [×3], C2×C22⋊C4 [×3], C22≀C2 [×12], C22≀C2 [×6], C22×D4 [×3], C22×D4 [×6], C2×C4○D4 [×3], 2+ (1+4) [×4], 2+ (1+4) [×6], C25, C2×C23⋊C4 [×3], C2≀C22 [×8], C2×C22≀C2 [×3], C2×2+ (1+4), C2×C2≀C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2≀C22 [×2], C2×C22≀C2, C2×C2≀C22
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >
►On 16 points - transitive group 16T245
(1 8)(2 7)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(1 11)(2 15)(3 13)(4 9)(5 12)(6 14)(7 10)(8 16)
(1 8)(2 5)(3 7)(4 6)(9 14)(10 13)(11 16)(12 15)
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 5)(3 7)(4 8)(10 12)(13 15)
G:=sub<Sym(16)| (1,8)(2,7)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,11)(2,15)(3,13)(4,9)(5,12)(6,14)(7,10)(8,16), (1,8)(2,5)(3,7)(4,6)(9,14)(10,13)(11,16)(12,15), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,5)(3,7)(4,8)(10,12)(13,15)>;
G:=Group( (1,8)(2,7)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,11)(2,15)(3,13)(4,9)(5,12)(6,14)(7,10)(8,16), (1,8)(2,5)(3,7)(4,6)(9,14)(10,13)(11,16)(12,15), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,5)(3,7)(4,8)(10,12)(13,15) );
G=PermutationGroup([(1,8),(2,7),(3,5),(4,6),(9,14),(10,15),(11,16),(12,13)], [(1,11),(2,15),(3,13),(4,9),(5,12),(6,14),(7,10),(8,16)], [(1,8),(2,5),(3,7),(4,6),(9,14),(10,13),(11,16),(12,15)], [(1,4),(2,3),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,6),(2,5),(3,7),(4,8),(10,12),(13,15)])
G:=TransitiveGroup(16,245);
►On 16 points - transitive group 16T246
(1 6)(2 5)(3 8)(4 7)(9 16)(10 13)(11 14)(12 15)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 16)(7 15)(8 14)
(1 3)(6 8)(9 11)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(10 12)(13 15)
G:=sub<Sym(16)| (1,6)(2,5)(3,8)(4,7)(9,16)(10,13)(11,14)(12,15), (1,9)(2,10)(3,11)(4,12)(5,13)(6,16)(7,15)(8,14), (1,3)(6,8)(9,11)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (10,12)(13,15)>;
G:=Group( (1,6)(2,5)(3,8)(4,7)(9,16)(10,13)(11,14)(12,15), (1,9)(2,10)(3,11)(4,12)(5,13)(6,16)(7,15)(8,14), (1,3)(6,8)(9,11)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (10,12)(13,15) );
G=PermutationGroup([(1,6),(2,5),(3,8),(4,7),(9,16),(10,13),(11,14),(12,15)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,16),(7,15),(8,14)], [(1,3),(6,8),(9,11),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(10,12),(13,15)])
G:=TransitiveGroup(16,246);
►On 16 points - transitive group 16T271
(1 2)(3 4)(5 7)(6 8)(9 14)(10 15)(11 16)(12 13)
(1 9)(2 14)(3 16)(4 11)(5 10)(6 12)(7 15)(8 13)
(1 5)(2 7)(3 8)(4 6)(9 10)(11 12)(13 16)(14 15)
(1 4)(2 3)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 2)(3 4)(5 7)(6 8)(9 13)(10 16)(11 15)(12 14)
G:=sub<Sym(16)| (1,2)(3,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13), (1,9)(2,14)(3,16)(4,11)(5,10)(6,12)(7,15)(8,13), (1,5)(2,7)(3,8)(4,6)(9,10)(11,12)(13,16)(14,15), (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,7)(6,8)(9,13)(10,16)(11,15)(12,14)>;
G:=Group( (1,2)(3,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13), (1,9)(2,14)(3,16)(4,11)(5,10)(6,12)(7,15)(8,13), (1,5)(2,7)(3,8)(4,6)(9,10)(11,12)(13,16)(14,15), (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,7)(6,8)(9,13)(10,16)(11,15)(12,14) );
G=PermutationGroup([(1,2),(3,4),(5,7),(6,8),(9,14),(10,15),(11,16),(12,13)], [(1,9),(2,14),(3,16),(4,11),(5,10),(6,12),(7,15),(8,13)], [(1,5),(2,7),(3,8),(4,6),(9,10),(11,12),(13,16),(14,15)], [(1,4),(2,3),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,2),(3,4),(5,7),(6,8),(9,13),(10,16),(11,15),(12,14)])
G:=TransitiveGroup(16,271);
Matrix representation ►G ⊆ 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | -1 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | -1 | -1 | 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 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | -1 | 0 | 0 | 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 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | -1 |
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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,-1,0,0,0,1,0,0,0,0,1,1,0,-1,0,0,0,2,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,-1,0,0,1,0,1,-1,0,0,0,0,1,0,0,0,0,0,2,-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,-1,0,0,0,1,0,0,0,0,0,1,-1,0,0,0,0,2,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,2,-1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | ··· | 2S | 4A | ··· | 4F | 4G | ··· | 4L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C2≀C22 |
| kernel | C2×C2≀C22 | C2×C23⋊C4 | C2≀C22 | C2×C22≀C2 | C2×2+ (1+4) | C22×C4 | C2×D4 | C24 | C2 |
| # reps | 1 | 3 | 8 | 3 | 1 | 3 | 6 | 3 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_2\wr C_2^2
% in TeX
G:=Group("C2xC2wrC2^2"); // GroupNames label
G:=SmallGroup(128,1755);
// by ID
G=gap.SmallGroup(128,1755);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,718,2028]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=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,e*b*e^-1=f*b*f=b*c*d,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations