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G = C22⋊Q16  order 64 = 26

The semidirect product of C22 and Q16 acting via Q16/Q8=C2

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

Aliases: Q8.6D4, C22⋊2Q16, C23.44D4, (C2×Q16)⋊1C2, (C2×C4).25D4, C4.23(C2×D4), C2.4(C2×Q16), Q8⋊C4⋊5C2, C4⋊C4.3C22, C22⋊C8.3C2, (C2×C8).2C22, C22⋊Q8.2C2, C2.12C22≀C2, (C2×C4).85C23, C22.81(C2×D4), (C2×Q8).3C22, (C22×Q8).6C2, C2.7(C8.C22), (C22×C4).46C22, SmallGroup(64,132)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C22⋊Q16
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C22⋊Q16
Lower central
C1 — C2 — C2×C4 — C22⋊Q16
Upper central
C1 — C22 — C22×C4 — C22⋊Q16
Jennings
C1 — C2 — C2 — C2×C4 — C22⋊Q16

Generators and relations for C22⋊Q16
 G = < a,b,c,d | a2=b2=c8=1, d2=c4, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 121 in 74 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, Q8, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C22⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C8.C22, C22⋊Q16

Character table of C22⋊Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111222244444884444
ρ11111111111111111111    trivial
ρ21111-1-111-11-11-11-11-1-11    linear of order 2
ρ311111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ41111-1-1111-11-1-11-1-111-1    linear of order 2
ρ51111-1-111-11-11-1-11-111-1    linear of order 2
ρ61111111111111-1-1-1-1-1-1    linear of order 2
ρ71111-1-1111-11-1-1-111-1-11    linear of order 2
ρ811111111-1-1-1-11-1-11111    linear of order 2
ρ92-22-200-2220-200000000    orthogonal lifted from D4
ρ102-22-2002-2020-20000000    orthogonal lifted from D4
ρ11222222-2-20000-2000000    orthogonal lifted from D4
ρ122-22-2002-20-2020000000    orthogonal lifted from D4
ρ132222-2-2-2-200002000000    orthogonal lifted from D4
ρ142-22-200-22-20200000000    orthogonal lifted from D4
ρ152-2-22-22000000000√2-√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ162-2-222-2000000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ172-2-22-22000000000-√2√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ182-2-222-2000000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C22⋊Q16
►On 32 points
Generators in S32
(1 5)(2 29)(3 7)(4 31)(6 25)(8 27)(9 13)(10 18)(11 15)(12 20)(14 22)(16 24)(17 21)(19 23)(26 30)(28 32)

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

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

C22⋊Q16 is a maximal subgroup of
 C23⋊Q16  C4⋊C4.6D4  C24.12D4  C24.103D4  C24.178D4  C24.106D4  Q8.(C2×D4)  (C2×Q8)⋊17D4  C42.226D4  C42.231D4  C42.235D4  C42.355C23  C42.361C23  C23⋊3Q16  C24.123D4  C24.128D4  C24.129D4  C4.162+ 1+4  C4.172+ 1+4  C42.269D4  C42.273D4  C42.276D4  C42.409C23  C42.411C23  SD16⋊6D4  SD16⋊8D4  Q16⋊9D4  SD16⋊3D4  Q16⋊4D4  SD16⋊10D4  D4×Q16  C42.465C23  C42.47C23  C42.48C23  C42.51C23  C42.476C23  C42.477C23  A4⋊2Q16  C23.14S4  Q8.1S4
 (C2×C2p)⋊Q16: (C2×C4)⋊Q16  C42.224D4  C42.267D4  Dic6.32D4  Dic6.37D4  (C2×C6)⋊8Q16  C22⋊Dic20  Dic10.37D4 ...
 D2p⋊Q16: D4⋊5Q16  D6⋊Q16  D6⋊5Q16  D10⋊4Q16  D10⋊5Q16  D14⋊4Q16  D14⋊5Q16 ...
C22⋊Q16 is a maximal quotient of
 C23⋊Q16  C24.17D4  C4⋊C4.20D4  Q8⋊Q16  Q8.Q16  D4.3Q16  Q8⋊3Q16  Q8⋊4Q16  C24.155D4  C23.37D8  Q8⋊(C4⋊C4)  C24.160D4  C23⋊2Q16  (C2×Q8)⋊Q8  C24.86D4  C4⋊C4.95D4  C4⋊C4⋊Q8
 (C2×C2p)⋊Q16: (C2×C4)⋊Q16  (C2×C4)⋊9Q16  (C2×C4)⋊2Q16  Dic6.32D4  Dic6.37D4  (C2×C6)⋊8Q16  C22⋊Dic20  Dic10.37D4 ...
 D2p⋊Q16: D4⋊Q16  D4⋊3Q16  D4⋊4Q16  D6⋊Q16  D6⋊5Q16  D10⋊4Q16  D10⋊5Q16  D14⋊4Q16 ...

Matrix representation of C22⋊Q16 ►in GL4(𝔽17) generated by

16000
01600
0010
00016
,
1000
0100
00160
00016
,
6600
14000
0001
00160
,
13000
4400
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[6,14,0,0,6,0,0,0,0,0,0,16,0,0,1,0],[13,4,0,0,0,4,0,0,0,0,1,0,0,0,0,16] >;
 

C22⋊Q16 in GAP, Magma, Sage, TeX 

C_2^2\rtimes Q_{16}
 
% in TeX
 
G:=Group("C2^2:Q16");
 
// GroupNames label
 
G:=SmallGroup(64,132);
 
// by ID
 
G=gap.SmallGroup(64,132);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,199,362,963,489,117]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^2=c^4,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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Character table of C22⋊Q16 in TeX

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