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G = C22⋊Q8  order 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⋊C4⋊3C2, (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

Derived series
C1 — C22 — C22⋊Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22⋊Q8
Lower central
C1 — C22 — C22⋊Q8
Upper central
C1 — C22 — C22⋊Q8
Jennings
C1 — C22 — 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 >

C1
C2
C2
C2
2C2
2C2
C4
C4
C22
C22
C22
2C4
2C4
2C4
2C4
2C4
2C22
2C22
C2×C4
C2×C4
C2×C4
C2×C4
C2×C4
C23
C2×C4
2Q8
2C2×C4
2C2×C4
2Q8
C2×Q8
C4⋊C4
C4⋊C4
C4⋊C4
C22⋊C4
C22⋊C4
C22×C4
C22⋊Q8

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 15)(6 16)(7 13)(8 14)

(1 9)(2 10)(3 11)(4 12)(5 15)(6 16)(7 13)(8 14)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 16 3 14)(2 15 4 13)(5 12 7 10)(6 11 8 9)
 
G:=sub<Sym(16)| (5,15)(6,16)(7,13)(8,14), (1,9)(2,10)(3,11)(4,12)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9)>;
 
G:=Group( (5,15)(6,16)(7,13)(8,14), (1,9)(2,10)(3,11)(4,12)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,3,14)(2,15,4,13)(5,12,7,10)(6,11,8,9) );
 
G=PermutationGroup([[(5,15),(6,16),(7,13),(8,14)], [(1,9),(2,10),(3,11),(4,12),(5,15),(6,16),(7,13),(8,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,3,14),(2,15,4,13),(5,12,7,10),(6,11,8,9)]])
 
G:=TransitiveGroup(16,31);
 

C22⋊Q8 is a maximal subgroup of
 C22.19C24  C23.36C23  C23.37C23  C23.38C23  C22.31C24  C22.32C24  C22.33C24  C22.35C24  C22.36C24  C23⋊2Q8  C23.41C23  D4⋊5D4  D4⋊6D4  Q8⋊5D4  D4×Q8  C22.45C24  C22.46C24  C22.50C24  C22.56C24  C22.57C24  S32⋊Q8  C62⋊Q8
 C4p.D4: C8.18D4  C8.D4  C4.D12  C12.48D4  D6⋊3Q8  D10⋊2Q8  C20.48D4  D10⋊3Q8 ...
 D2p⋊Q8: D4⋊3Q8  D6⋊Q8  D10⋊Q8  D14⋊Q8  D22⋊Q8  D26⋊Q8 ...
 C23.D2p: C23.31D4  C22⋊SD16  C22⋊Q16  D4.7D4  C8⋊8D4  C8⋊D4  C23.47D4  C23.48D4 ...
C22⋊Q8 is a maximal quotient of
 C23.63C23  C23.65C23  C23.67C23  C23⋊Q8  C23.78C23  C23.Q8  C23.81C23  C23.4Q8  C23.83C23  Q8⋊Q8  C4.Q16  D4.Q8  Q8.Q8  S32⋊Q8  C62⋊Q8
 D2p⋊Q8: D4⋊Q8  D4⋊2Q8  D6⋊Q8  C4.D12  D6⋊3Q8  D10⋊Q8  D10⋊2Q8  D10⋊3Q8 ...
 C23.D2p: C23.7Q8  C23.8Q8  Dic3.D4  C12.48D4  Dic5.14D4  C20.48D4  C22⋊Dic14  C28.48D4 ...

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
 

Export

Subgroup lattice of C22⋊Q8 in TeX
Character table of C22⋊Q8 in TeX

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