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G = (C2×C4)≀C2order 128 = 27

Wreath product of C2×C4 by C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C4)≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — (C2×C4)≀C2
 Lower central C1 — C2 — C2×C4 — (C2×C4)≀C2
 Upper central C1 — C2×C4 — C23×C4 — (C2×C4)≀C2
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)≀C2

Generators and relations for (C2×C4)≀C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 420 in 221 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×4], C4 [×13], C22 [×3], C22 [×4], C22 [×17], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×43], D4 [×8], Q8 [×2], C23, C23 [×2], C23 [×7], C42 [×4], C42 [×7], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×19], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×2], C4≀C2 [×4], C2×C42 [×2], C2×C42 [×5], C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C23×C4, C23×C4, C2×C4○D4, C426C4 [×2], C24.4C4, C2×C4≀C2 [×2], C22×C42, C22.19C24, (C2×C4)≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C2×C4≀C2 [×2], (C2×C4)≀C2

Permutation representations of (C2×C4)≀C2
On 16 points - transitive group 16T211
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 5 4 7)(2 6 3 8)(9 13 11 15)(10 14 12 16)
(1 12 3 15)(2 13 4 10)(5 16 8 9)(6 11 7 14)
(1 10)(2 15)(3 13)(4 12)(5 14)(6 9)(7 16)(8 11)

G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,4,7)(2,6,3,8)(9,13,11,15)(10,14,12,16), (1,12,3,15)(2,13,4,10)(5,16,8,9)(6,11,7,14), (1,10)(2,15)(3,13)(4,12)(5,14)(6,9)(7,16)(8,11)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,4,7)(2,6,3,8)(9,13,11,15)(10,14,12,16), (1,12,3,15)(2,13,4,10)(5,16,8,9)(6,11,7,14), (1,10)(2,15)(3,13)(4,12)(5,14)(6,9)(7,16)(8,11) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,5,4,7),(2,6,3,8),(9,13,11,15),(10,14,12,16)], [(1,12,3,15),(2,13,4,10),(5,16,8,9),(6,11,7,14)], [(1,10),(2,15),(3,13),(4,12),(5,14),(6,9),(7,16),(8,11)])

G:=TransitiveGroup(16,211);

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 4D 4E ··· 4Z 4AA 4AB 4AC 8A 8B 8C 8D order 1 2 2 2 2 ··· 2 2 4 4 4 4 4 ··· 4 4 4 4 8 8 8 8 size 1 1 1 1 2 ··· 2 8 1 1 1 1 2 ··· 2 8 8 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 C4○D4 C4≀C2 kernel (C2×C4)≀C2 C42⋊6C4 C24.4C4 C2×C4≀C2 C22×C42 C22.19C24 C4⋊D4 C22⋊Q8 C42 C22×C4 C24 C2×C4 C22 # reps 1 2 1 2 1 1 4 4 4 3 1 4 16

Matrix representation of (C2×C4)≀C2 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 13 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 0 16 0 0 1 0 0 0 0 0 0 4 0 0 13 0
,
 0 16 0 0 16 0 0 0 0 0 0 13 0 0 4 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0],[0,16,0,0,16,0,0,0,0,0,0,4,0,0,13,0] >;

(C2×C4)≀C2 in GAP, Magma, Sage, TeX

(C_2\times C_4)\wr C_2
% in TeX

G:=Group("(C2xC4)wrC2");
// GroupNames label

G:=SmallGroup(128,628);
// by ID

G=gap.SmallGroup(128,628);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,248,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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