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## G = C22⋊SD16order 64 = 26

### The semidirect product of C22 and SD16 acting via SD16/D4=C2

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

Aliases: D4.6D4, C222SD16, C23.43D4, C4⋊C42C22, C22⋊C89C2, (C2×C8)⋊6C22, (C2×C4).24D4, C4.22(C2×D4), C22⋊Q81C2, D4⋊C49C2, (C2×SD16)⋊9C2, (C2×Q8)⋊1C22, C2.6(C2×SD16), C2.11C22≀C2, C2.8(C8⋊C22), (C2×C4).84C23, (C22×D4).8C2, C22.80(C2×D4), (C2×D4).54C22, (C22×C4).45C22, 2-Sylow(CO-(4,3)), SmallGroup(64,131)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C22⋊SD16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C22⋊SD16
 Lower central C1 — C2 — C2×C4 — C22⋊SD16
 Upper central C1 — C22 — C22×C4 — C22⋊SD16
 Jennings C1 — C2 — C2 — C2×C4 — C22⋊SD16

Generators and relations for C22⋊SD16
G = < a,b,c,d | a2=b2=c8=d2=1, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 201 in 94 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C22⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16

Character table of C22⋊SD16

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 2 2 4 8 8 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ9 2 -2 2 -2 0 0 -2 0 2 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 2 0 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 0 0 0 -2 0 2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 0 0 2 0 -2 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Permutation representations of C22⋊SD16
On 16 points - transitive group 16T155
Generators in S16
(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11)(2 14)(3 9)(4 12)(5 15)(6 10)(7 13)(8 16)

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

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

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

G:=TransitiveGroup(16,155);

C22⋊SD16 is a maximal subgroup of
C24.103D4  C24.177D4  C24.106D4  (C2×D4)⋊21D4  C42.225D4  C42.228D4  C42.232D4  C42.352C23  C42.357C23  C234SD16  C24.126D4  C42.269D4  C42.273D4  A4⋊SD16
D2p⋊SD16: D47SD16  D65SD16  D66SD16  D20.8D4  D106SD16  D4.6D28  D146SD16 ...
(Cp×D4).D4: C4⋊C4.D4  D4.(C2×D4)  C4.2+ 1+4  C4.152+ 1+4  D89D4  SD16⋊D4  SD166D4  D810D4 ...
C4⋊C4⋊D2p: C23⋊SD16  C24.9D4  D12.36D4  D20.36D4  D28.36D4 ...
(C2×C2p)⋊SD16: (C2×C4)⋊SD16  C42.222D4  C42.266D4  D12.31D4  D20.31D4  D28.31D4 ...
C8pD4⋊C2: C24.121D4  C24.127D4  C42.275D4  C42.408C23  C42.410C23 ...
C22⋊SD16 is a maximal quotient of
D2p⋊SD16: D42SD16  D4.D8  D44SD16  D65SD16  D66SD16  D20.8D4  D106SD16  D4.6D28 ...
(Cp×D4).D4: C232SD16  Q8⋊D4⋊C2  C24.16D4  C4⋊C4.19D4  Q82SD16  D4.SD16  D4.3Q16  Q83SD16 ...
(C22×C2p).D4: C24.159D4  C24.160D4  (C2×Q8)⋊Q8  (C2×C8)⋊Q8  D12.31D4  D12.36D4  D20.31D4  D20.36D4 ...

Matrix representation of C22⋊SD16 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 0 16 0 0 16 0 0 0 0 0 0 7 0 0 5 7
,
 16 0 0 0 0 16 0 0 0 0 1 15 0 0 0 16
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,16,0,0,0,0,0,0,5,0,0,7,7],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,15,16] >;

C22⋊SD16 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C2^2:SD16");
// GroupNames label

G:=SmallGroup(64,131);
// by ID

G=gap.SmallGroup(64,131);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,963,489,117]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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