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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, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4≀C2, C2×C42, C2×C42, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C2×M4(2), C23×C4, C23×C4, C2×C4○D4, C426C4, C24.4C4, C2×C4≀C2, C22×C42, C22.19C24, (C2×C4)≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4≀C2, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C2×C4≀C2, (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 4 8 6)(2 3 7 5)(9 14 11 16)(10 15 12 13)
(1 10 7 14)(2 16 8 12)(3 9 6 13)(4 15 5 11)
(1 12)(2 14)(3 11)(4 13)(5 9)(6 15)(7 16)(8 10)

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

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

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

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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