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## G = C2×C4○D4order 32 = 25

### Direct product of C2 and C4○D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4○D4, D43C22, C2.3C24, C4.8C23, Q83C22, C22.1C23, C23.11C22, C4(C2×D4), (C2×C4)D4, C4(C2×Q8), (C2×C4)Q8, C4(C4○D4), (C2×D4)⋊7C2, (C2×Q8)⋊6C2, (C22×C4)⋊6C2, (C2×C4)⋊5C22, (C2×C4)(C2×Q8), SmallGroup(32,48)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C4○D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4
 Lower central C1 — C2 — C2×C4○D4
 Upper central C1 — C2×C4 — C2×C4○D4
 Jennings C1 — C2 — C2×C4○D4

Generators and relations for C2×C4○D4
G = < a,b,c,d | a2=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 94 in 82 conjugacy classes, 70 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4○D4
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4

Character table of C2×C4○D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ6 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 linear of order 2 ρ7 1 -1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ8 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ10 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ11 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ12 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ13 1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 linear of order 2 ρ14 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ15 1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 linear of order 2 ρ16 1 -1 1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ17 2 2 -2 -2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 -2 -2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4

Permutation representations of C2×C4○D4
On 16 points - transitive group 16T18
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4 3 2)(5 6 7 8)(9 12 11 10)(13 14 15 16)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)

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

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

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

G:=TransitiveGroup(16,18);

Matrix representation of C2×C4○D4 in GL3(𝔽5) generated by

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

C2×C4○D4 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_4
% in TeX

G:=Group("C2xC4oD4");
// GroupNames label

G:=SmallGroup(32,48);
// by ID

G=gap.SmallGroup(32,48);
# by ID

G:=PCGroup([5,-2,2,2,2,-2,181,72]);
// Polycyclic

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

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