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## G = (C2×C8)⋊D4order 128 = 27

### 3rd semidirect product of C2×C8 and D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — (C2×C8)⋊D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — (C2×C8)⋊D4
 Lower central C1 — C2 — C23 — (C2×C8)⋊D4
 Upper central C1 — C4 — C22×C4 — (C2×C8)⋊D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8)⋊D4

Generators and relations for (C2×C8)⋊D4
G = < a,b,c,d | a2=b8=c4=d2=1, cbc-1=dbd=ab=ba, cac-1=ab4, ad=da, dcd=c-1 >

Subgroups: 340 in 171 conjugacy classes, 54 normal (20 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×15], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×13], D4 [×10], Q8 [×2], C23, C23 [×2], C23 [×5], C42, C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×6], M4(2) [×12], C22×C4 [×2], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×4], C4.D4, C4.10D4, C42⋊C2, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C22⋊Q8, C22.D4 [×2], C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C23×C4, C2×C4○D4, M4(2)⋊4C4 [×2], C24.4C4 [×2], M4(2).8C22, C22.19C24, Q8○M4(2), (C2×C8)⋊D4
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], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, (C2×C8)⋊D4

Permutation representations of (C2×C8)⋊D4
On 16 points - transitive group 16T209
Generators in S16
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 7 5 3)(2 16)(4 10)(6 12)(8 14)(9 11 13 15)
(1 9)(2 6)(3 11)(4 8)(5 13)(7 15)(10 14)(12 16)

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

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

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

G:=TransitiveGroup(16,209);

On 16 points - transitive group 16T279
Generators in S16
(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11 5 15)(2 12)(3 13 7 9)(4 14)(6 16)(8 10)
(9 13)(11 15)

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

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

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

G:=TransitiveGroup(16,279);

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 size 1 1 2 2 2 4 4 4 4 1 1 2 2 2 4 4 4 4 8 8 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 D4 C4○D4 (C2×C8)⋊D4 kernel (C2×C8)⋊D4 M4(2)⋊4C4 C24.4C4 M4(2).8C22 C22.19C24 Q8○M4(2) C22≀C2 C22.D4 C2×C8 C22×C4 C2×D4 C2×Q8 C2×C4 C1 # reps 1 2 2 1 1 1 4 4 4 2 1 1 4 4

Matrix representation of (C2×C8)⋊D4 in GL4(𝔽5) generated by

 0 4 3 3 0 3 1 0 0 2 2 0 2 4 0 0
,
 0 3 4 2 1 0 0 4 0 0 3 1 0 0 4 2
,
 0 1 2 0 2 3 4 0 4 1 1 4 0 1 2 1
,
 0 0 1 3 0 1 0 0 0 0 1 0 2 0 3 0
G:=sub<GL(4,GF(5))| [0,0,0,2,4,3,2,4,3,1,2,0,3,0,0,0],[0,1,0,0,3,0,0,0,4,0,3,4,2,4,1,2],[0,2,4,0,1,3,1,1,2,4,1,2,0,0,4,1],[0,0,0,2,0,1,0,0,1,0,1,3,3,0,0,0] >;

(C2×C8)⋊D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes D_4
% in TeX

G:=Group("(C2xC8):D4");
// GroupNames label

G:=SmallGroup(128,623);
// by ID

G=gap.SmallGroup(128,623);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,521,2804,1411,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,d*c*d=c^-1>;
// generators/relations

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