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G = C22⋊Q16order 64 = 26

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

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

Aliases: Q8.6D4, C222Q16, C23.44D4, (C2×Q16)⋊1C2, (C2×C4).25D4, C4.23(C2×D4), C2.4(C2×Q16), Q8⋊C45C2, 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

C1C2×C4 — C22⋊Q16
C1C2C22C2×C4C22×C4C22×Q8 — C22⋊Q16
C1C2C2×C4 — C22⋊Q16
C1C22C22×C4 — C22⋊Q16
C1C2C2C2×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 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×4], Q8 [×8], C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C22⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, Q16 [×2], C2×D4 [×3], 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
ρ152-2-22-220000000002-22-2    symplectic lifted from Q16, Schur index 2
ρ162-2-222-200000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ172-2-22-22000000000-22-22    symplectic lifted from Q16, Schur index 2
ρ182-2-222-2000000000-2-222    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  C233Q16  C24.123D4  C24.128D4  C24.129D4  C4.162+ 1+4  C4.172+ 1+4  C42.269D4  C42.273D4  C42.276D4  C42.409C23  C42.411C23  SD166D4  SD168D4  Q169D4  SD163D4  Q164D4  SD1610D4  D4×Q16  C42.465C23  C42.47C23  C42.48C23  C42.51C23  C42.476C23  C42.477C23  A42Q16  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: D45Q16  D6⋊Q16  D65Q16  D104Q16  D105Q16  D144Q16  D145Q16 ...
C22⋊Q16 is a maximal quotient of
C23⋊Q16  C24.17D4  C4⋊C4.20D4  Q8⋊Q16  Q8.Q16  D4.3Q16  Q83Q16  Q84Q16  C24.155D4  C23.37D8  Q8⋊(C4⋊C4)  C24.160D4  C232Q16  (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  D43Q16  D44Q16  D6⋊Q16  D65Q16  D104Q16  D105Q16  D144Q16 ...

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

Export

Character table of C22⋊Q16 in TeX

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