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G = C4⋊C8order 32 = 25

The semidirect product of C4 and C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: C4⋊C8, C4.4Q8, C4.18D4, C42.2C2, C2.3M4(2), (C2×C4).4C4, C2.2(C2×C8), (C2×C8).2C2, C2.2(C4⋊C4), C22.9(C2×C4), (C2×C4).33C22, SmallGroup(32,12)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4⋊C8
C1C2C4C2×C4C42 — C4⋊C8
C1C2 — C4⋊C8
C1C2×C4 — C4⋊C8
C1C2C2C2×C4 — C4⋊C8

Generators and relations for C4⋊C8
 G = < a,b | a4=b8=1, bab-1=a-1 >

2C4
2C8
2C8

Character table of C4⋊C8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ51111-1-1-1-11-11-1-ii-ii-iii-i    linear of order 4
ρ61111-1-1-1-1-11-11i-ii-i-iii-i    linear of order 4
ρ71111-1-1-1-11-11-1i-ii-ii-i-ii    linear of order 4
ρ81111-1-1-1-1-11-11-ii-iii-i-ii    linear of order 4
ρ91-11-1-i-iii-i-1i1ζ85ζ87ζ8ζ83ζ87ζ85ζ8ζ83    linear of order 8
ρ101-11-1-i-iiii1-i-1ζ8ζ83ζ85ζ87ζ87ζ85ζ8ζ83    linear of order 8
ρ111-11-1-i-iii-i-1i1ζ8ζ83ζ85ζ87ζ83ζ8ζ85ζ87    linear of order 8
ρ121-11-1-i-iiii1-i-1ζ85ζ87ζ8ζ83ζ83ζ8ζ85ζ87    linear of order 8
ρ131-11-1ii-i-i-i1i-1ζ83ζ8ζ87ζ85ζ85ζ87ζ83ζ8    linear of order 8
ρ141-11-1ii-i-i-i1i-1ζ87ζ85ζ83ζ8ζ8ζ83ζ87ζ85    linear of order 8
ρ151-11-1ii-i-ii-1-i1ζ87ζ85ζ83ζ8ζ85ζ87ζ83ζ8    linear of order 8
ρ161-11-1ii-i-ii-1-i1ζ83ζ8ζ87ζ85ζ8ζ83ζ87ζ85    linear of order 8
ρ1722-2-22-22-2000000000000    orthogonal lifted from D4
ρ1822-2-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ192-2-222i-2i-2i2i000000000000    complex lifted from M4(2)
ρ202-2-22-2i2i2i-2i000000000000    complex lifted from M4(2)

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

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

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

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

Matrix representation of C4⋊C8 in GL3(𝔽17) generated by

100
01615
011
,
200
0160
011
G:=sub<GL(3,GF(17))| [1,0,0,0,16,1,0,15,1],[2,0,0,0,16,1,0,0,1] >;

C4⋊C8 in GAP, Magma, Sage, TeX

C_4\rtimes C_8
% in TeX

G:=Group("C4:C8");
// GroupNames label

G:=SmallGroup(32,12);
// by ID

G=gap.SmallGroup(32,12);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,58]);
// Polycyclic

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

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