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## G = C2×C8.C4order 64 = 26

### Direct product of C2 and C8.C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8.C4
 Lower central C1 — C2 — C4 — C2×C8.C4
 Upper central C1 — C2×C4 — C22×C4 — C2×C8.C4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8.C4

Generators and relations for C2×C8.C4
G = < a,b,c | a2=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×C8.C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 8O 8P size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 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 -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 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 -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 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 -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 -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 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 -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 -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 -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 -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 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 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 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 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 -2 2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 2 2 2 2 2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 2 -2 -2 2 0 0 2i -2i 2i -2i 0 0 √2 -√-2 √-2 √-2 -√2 -√2 √2 -√-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ22 2 2 -2 -2 0 0 -2i -2i 2i 2i 0 0 √2 -√-2 -√-2 √-2 √2 -√2 -√2 √-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ23 2 -2 -2 2 0 0 2i -2i 2i -2i 0 0 -√2 √-2 -√-2 -√-2 √2 √2 -√2 √-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ24 2 2 -2 -2 0 0 -2i -2i 2i 2i 0 0 -√2 √-2 √-2 -√-2 -√2 √2 √2 -√-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ25 2 -2 -2 2 0 0 -2i 2i -2i 2i 0 0 √2 √-2 -√-2 -√-2 -√2 -√2 √2 √-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ26 2 2 -2 -2 0 0 2i 2i -2i -2i 0 0 √2 √-2 √-2 -√-2 √2 -√2 -√2 -√-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ27 2 -2 -2 2 0 0 -2i 2i -2i 2i 0 0 -√2 -√-2 √-2 √-2 √2 √2 -√2 -√-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ28 2 2 -2 -2 0 0 2i 2i -2i -2i 0 0 -√2 -√-2 -√-2 √-2 -√2 √2 √2 √-2 0 0 0 0 0 0 0 0 complex lifted from C8.C4

Smallest permutation representation of C2×C8.C4
On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 32 3 30 5 28 7 26)(2 31 4 29 6 27 8 25)(9 18 15 20 13 22 11 24)(10 17 16 19 14 21 12 23)

G:=sub<Sym(32)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,32,3,30,5,28,7,26)(2,31,4,29,6,27,8,25)(9,18,15,20,13,22,11,24)(10,17,16,19,14,21,12,23)>;

G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,32,3,30,5,28,7,26)(2,31,4,29,6,27,8,25)(9,18,15,20,13,22,11,24)(10,17,16,19,14,21,12,23) );

G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,32,3,30,5,28,7,26),(2,31,4,29,6,27,8,25),(9,18,15,20,13,22,11,24),(10,17,16,19,14,21,12,23)]])

Matrix representation of C2×C8.C4 in GL3(𝔽17) generated by

 16 0 0 0 1 0 0 0 1
,
 1 0 0 0 9 0 0 5 2
,
 13 0 0 0 1 2 0 6 16
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,9,5,0,0,2],[13,0,0,0,1,6,0,2,16] >;

C2×C8.C4 in GAP, Magma, Sage, TeX

C_2\times C_8.C_4
% in TeX

G:=Group("C2xC8.C4");
// GroupNames label

G:=SmallGroup(64,110);
// by ID

G=gap.SmallGroup(64,110);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,963,117,88]);
// Polycyclic

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

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