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G = C4⋊Q8order 32 = 25

The semidirect product of C4 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C4⋊Q8, C4.5D4, C42.5C2, C22.18C23, C4⋊C4.5C2, C2.5(C2×Q8), C2.10(C2×D4), (C2×Q8).3C2, (C2×C4).5C22, SmallGroup(32,35)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊Q8
C1C2C22C2×C4C42 — C4⋊Q8
C1C22 — C4⋊Q8
C1C22 — C4⋊Q8
C1C22 — C4⋊Q8

Generators and relations for C4⋊Q8
 G = < a,b,c | a4=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

2C4
2C4
2C4
2C4
2Q8
2Q8
2Q8
2Q8

Character table of C4⋊Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J
 size 11112222224444
ρ111111111111111    trivial
ρ21111-1-1-11-1111-1-1    linear of order 2
ρ311111-11-1-1-11-1-11    linear of order 2
ρ41111-11-1-11-11-11-1    linear of order 2
ρ51111-11-1-11-1-11-11    linear of order 2
ρ611111-11-1-1-1-111-1    linear of order 2
ρ71111-1-1-11-11-1-111    linear of order 2
ρ81111111111-1-1-1-1    linear of order 2
ρ92-22-20200-200000    orthogonal lifted from D4
ρ102-22-20-200200000    orthogonal lifted from D4
ρ1122-2-2000-2020000    symplectic lifted from Q8, Schur index 2
ρ122-2-2220-20000000    symplectic lifted from Q8, Schur index 2
ρ1322-2-200020-20000    symplectic lifted from Q8, Schur index 2
ρ142-2-22-2020000000    symplectic lifted from Q8, Schur index 2

Smallest permutation representation of C4⋊Q8
Regular action on 32 points
Generators in S32
(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 7 22 10)(2 8 23 11)(3 5 24 12)(4 6 21 9)(13 18 25 29)(14 19 26 30)(15 20 27 31)(16 17 28 32)
(1 14 22 26)(2 13 23 25)(3 16 24 28)(4 15 21 27)(5 32 12 17)(6 31 9 20)(7 30 10 19)(8 29 11 18)

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

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

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

Matrix representation of C4⋊Q8 in GL4(𝔽5) generated by

1000
0100
0001
0040
,
3000
0200
0040
0004
,
0100
4000
0040
0001
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[3,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,1] >;

C4⋊Q8 in GAP, Magma, Sage, TeX

C_4\rtimes Q_8
% in TeX

G:=Group("C4:Q8");
// GroupNames label

G:=SmallGroup(32,35);
// by ID

G=gap.SmallGroup(32,35);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,40,101,46,302,72]);
// Polycyclic

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

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