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## G = C2×M4(2)  order 32 = 25

### Direct product of C2 and M4(2)

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

Aliases: C2×M4(2), C84C22, C4M4(2), C23.3C4, C4.11C23, (C2×C8)⋊6C2, (C2×C4).6C4, C4.10(C2×C4), (C22×C4).6C2, C2.6(C22×C4), (C2×C4).24C22, C22.11(C2×C4), SmallGroup(32,37)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2)
 Lower central C1 — C2 — C2×M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)
 Jennings C1 — C2 — C2 — C4 — C2×M4(2)

Generators and relations for C2×M4(2)
G = < a,b,c | a2=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Character table of C2×M4(2)

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 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 i -i -i -i i i i -i linear of order 4 ρ10 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i i -i -i i i -i i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -i i i i -i -i -i i linear of order 4 ρ12 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 i -i i i -i -i i -i linear of order 4 ρ13 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 -i i i -i -i i i -i linear of order 4 ρ14 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 i -i i -i -i i -i i linear of order 4 ρ15 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 i -i -i i i -i -i i linear of order 4 ρ16 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 -i i -i i i -i i -i linear of order 4 ρ17 2 2 -2 -2 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ18 2 2 -2 -2 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 -2 2 0 0 2i -2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 -2 2 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)

Permutation representations of C2×M4(2)
On 16 points - transitive group 16T15
Generators in S16
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)

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

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

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

G:=TransitiveGroup(16,15);

Matrix representation of C2×M4(2) in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 0 1 0 4 0
,
 16 0 0 0 1 0 0 0 16
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,0,4,0,1,0],[16,0,0,0,1,0,0,0,16] >;

C2×M4(2) in GAP, Magma, Sage, TeX

C_2\times M_4(2)
% in TeX

G:=Group("C2xM4(2)");
// GroupNames label

G:=SmallGroup(32,37);
// by ID

G=gap.SmallGroup(32,37);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,40,181,58]);
// Polycyclic

G:=Group<a,b,c|a^2=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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