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G = C22⋊Q8order 32 = 25

The semidirect product of C22 and Q8 acting via Q8/C4=C2

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

Aliases: C22⋊Q8, C4.13D4, C23.9C22, C22.12C23, C4⋊C43C2, (C2×Q8)⋊1C2, C2.6(C2×D4), C2.3(C2×Q8), C2.5(C4○D4), C22⋊C4.1C2, (C22×C4).5C2, (C2×C4).21C22, SmallGroup(32,29)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22⋊Q8
C1C2C22C23C22×C4 — C22⋊Q8
C1C22 — C22⋊Q8
C1C22 — C22⋊Q8
C1C22 — C22⋊Q8

Generators and relations for C22⋊Q8
 G = < a,b,c,d | a2=b2=c4=1, d2=c2, dad-1=ab=ba, ac=ca, bc=cb, bd=db, dcd-1=c-1 >

2C2
2C2
2C4
2C4
2C4
2C4
2C4
2C22
2C22
2Q8
2C2×C4
2C2×C4
2Q8

Character table of C22⋊Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H
 size 11112222224444
ρ111111111111111    trivial
ρ21111-1-1-1-111-11-11    linear of order 2
ρ31111-1-111-1-1-111-1    linear of order 2
ρ4111111-1-1-1-111-1-1    linear of order 2
ρ51111-1-1-1-1111-11-1    linear of order 2
ρ61111111111-1-1-1-1    linear of order 2
ρ7111111-1-1-1-1-1-111    linear of order 2
ρ81111-1-111-1-11-1-11    linear of order 2
ρ922-2-200002-20000    orthogonal lifted from D4
ρ1022-2-20000-220000    orthogonal lifted from D4
ρ112-22-22-200000000    symplectic lifted from Q8, Schur index 2
ρ122-22-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ132-2-2200-2i2i000000    complex lifted from C4○D4
ρ142-2-22002i-2i000000    complex lifted from C4○D4

Permutation representations of C22⋊Q8
On 16 points - transitive group 16T31
Generators in S16
(5 13)(6 14)(7 15)(8 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14 3 16)(2 13 4 15)(5 12 7 10)(6 11 8 9)

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

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

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

G:=TransitiveGroup(16,31);

Matrix representation of C22⋊Q8 in GL4(𝔽5) generated by

4000
0400
0040
0001
,
1000
0100
0040
0004
,
0100
4000
0020
0003
,
0200
2000
0001
0040
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,1,0,0,0,0,0,2,0,0,0,0,3],[0,2,0,0,2,0,0,0,0,0,0,4,0,0,1,0] >;

C22⋊Q8 in GAP, Magma, Sage, TeX

C_2^2\rtimes Q_8
% in TeX

G:=Group("C2^2:Q8");
// GroupNames label

G:=SmallGroup(32,29);
// by ID

G=gap.SmallGroup(32,29);
# by ID

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

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

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