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G = C22:SD16  order 64 = 26

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

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

Aliases: D4.6D4, C22:2SD16, C23.43D4, C4:C4:2C22, C22:C8:9C2, (C2xC8):6C22, (C2xC4).24D4, C4.22(C2xD4), C22:Q8:1C2, D4:C4:9C2, (C2xSD16):9C2, (C2xQ8):1C22, C2.6(C2xSD16), C2.11C22wrC2, C2.8(C8:C22), (C2xC4).84C23, (C22xD4).8C2, C22.80(C2xD4), (C2xD4).54C22, (C22xC4).45C22, 2-Sylow(CO-(4,3)), SmallGroup(64,131)

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

Derived series
C1 — C2xC4 — C22:SD16
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C22xD4 — C22:SD16
Lower central
C1 — C2 — C2xC4 — C22:SD16
Upper central
C1 — C22 — C22xC4 — C22:SD16
Jennings
C1 — C2 — C2 — C2xC4 — 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, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C22:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C24, C22:C8, D4:C4, C22:Q8, C2xSD16, C22xD4, C22:SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C22wrC2, C2xSD16, C8:C22, C22:SD16

Character table of C22:SD16

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E8A8B8C8D
 size 1111224444224884444
ρ11111111111111111111    trivial
ρ21111-1-11-11-111-11-1-1-111    linear of order 2
ρ3111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ41111-1-1-11-1111-11-111-1-1    linear of order 2
ρ51111111111111-1-1-1-1-1-1    linear of order 2
ρ61111-1-11-11-111-1-1111-1-1    linear of order 2
ρ7111111-1-1-1-1111-1-11111    linear of order 2
ρ81111-1-1-11-1111-1-11-1-111    linear of order 2
ρ92-22-200-20202-20000000    orthogonal lifted from D4
ρ102222220000-2-2-2000000    orthogonal lifted from D4
ρ112-22-200020-2-220000000    orthogonal lifted from D4
ρ122222-2-20000-2-22000000    orthogonal lifted from D4
ρ132-22-2000-202-220000000    orthogonal lifted from D4
ρ142-22-20020-202-20000000    orthogonal lifted from D4
ρ152-2-22-22000000000√-2-√-2√-2-√-2    complex lifted from SD16
ρ162-2-222-2000000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ172-2-222-2000000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ182-2-22-22000000000-√-2√-2-√-2√-2    complex lifted from SD16
ρ1944-4-4000000000000000    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  (C2xD4):21D4  C42.225D4  C42.228D4  C42.232D4  C42.352C23  C42.357C23  C23:4SD16  C24.126D4  C42.269D4  C42.273D4  A4:SD16
 D2p:SD16: D4:7SD16  D6:5SD16  D6:6SD16  D20.8D4  D10:6SD16  D4.6D28  D14:6SD16 ...
 (CpxD4).D4: C4:C4.D4  D4.(C2xD4)  C4.2+ 1+4  C4.152+ 1+4  D8:9D4  SD16:D4  SD16:6D4  D8:10D4 ...
 C4:C4:D2p: C23:SD16  C24.9D4  D12.36D4  D20.36D4  D28.36D4 ...
 (C2xC2p):SD16: (C2xC4):SD16  C42.222D4  C42.266D4  D12.31D4  D20.31D4  D28.31D4 ...
 C8:pD4:C2: C24.121D4  C24.127D4  C42.275D4  C42.408C23  C42.410C23 ...
C22:SD16 is a maximal quotient of
 D2p:SD16: D4:2SD16  D4.D8  D4:4SD16  D6:5SD16  D6:6SD16  D20.8D4  D10:6SD16  D4.6D28 ...
 (CpxD4).D4: C23:2SD16  Q8:D4:C2  C24.16D4  C4:C4.19D4  Q8:2SD16  D4.SD16  D4.3Q16  Q8:3SD16 ...
 (C22xC2p).D4: C24.159D4  C24.160D4  (C2xQ8):Q8  (C2xC8):Q8  D12.31D4  D12.36D4  D20.31D4  D20.36D4 ...

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

1000
01600
0010
0001
,
16000
01600
0010
0001
,
01600
16000
0007
0057
,
16000
01600
00115
00016
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
 

Export

Character table of C22:SD16 in TeX

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