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G = C22.SD16order 64 = 26

1st non-split extension by C22 of SD16 acting via SD16/Q8=C2

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

Aliases: C22.2D8, C23.30D4, C22.7SD16, C4⋊C41C4, C2.4C4≀C2, (C2×D4)⋊1C4, C22⋊C82C2, (C2×C4).94D4, C4⋊D4.1C2, C2.4(C23⋊C4), C2.3(D4⋊C4), C2.C426C2, (C22×C4).16C22, C22.35(C22⋊C4), (C2×C4).8(C2×C4), SmallGroup(64,8)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C22.SD16
C1C2C22C23C22×C4C4⋊D4 — C22.SD16
C1C22C2×C4 — C22.SD16
C1C22C22×C4 — C22.SD16
C1C2C22C22×C4 — C22.SD16

Generators and relations for C22.SD16
 G = < a,b,c,d | a2=b2=c8=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=abc3 >

2C2
2C2
8C2
2C4
2C4
2C22
2C22
4C22
4C4
4C22
4C22
4C4
4C4
2C2×C4
2C23
2C2×C4
2C2×C4
4C8
4D4
4D4
4C2×C4
4C2×C4
4D4
4C2×C4
2C22⋊C4
2C2×C8
2C22×C4
2C2×D4

Character table of C22.SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111228224444484444
ρ11111111111111111111    trivial
ρ2111111-11111111-1-1-1-1-1    linear of order 2
ρ3111111-111-1-11-1-1-11111    linear of order 2
ρ4111111111-1-11-1-11-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-1-ii1i-i1ii-i-i    linear of order 4
ρ61111-1-11-1-1-ii1i-i-1-i-iii    linear of order 4
ρ71111-1-11-1-1i-i1-ii-1ii-i-i    linear of order 4
ρ81111-1-1-1-1-1i-i1-ii1-i-iii    linear of order 4
ρ92222220-2-200-20000000    orthogonal lifted from D4
ρ102222-2-202200-20000000    orthogonal lifted from D4
ρ112-2-222-2000000000-22-22    orthogonal lifted from D8
ρ122-2-222-20000000002-22-2    orthogonal lifted from D8
ρ132-22-2000-2i2i-1+i-1-i01+i1-i00000    complex lifted from C4≀C2
ρ142-22-20002i-2i1+i1-i0-1+i-1-i00000    complex lifted from C4≀C2
ρ152-2-22-22000000000--2-2-2--2    complex lifted from SD16
ρ162-22-20002i-2i-1-i-1+i01-i1+i00000    complex lifted from C4≀C2
ρ172-22-2000-2i2i1-i1+i0-1-i-1+i00000    complex lifted from C4≀C2
ρ182-2-22-22000000000-2--2--2-2    complex lifted from SD16
ρ1944-4-4000000000000000    orthogonal lifted from C23⋊C4

Permutation representations of C22.SD16
On 16 points - transitive group 16T163
Generators in S16
(1 5)(2 12)(3 7)(4 14)(6 16)(8 10)(9 13)(11 15)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 6)(2 5)(3 10)(4 9)(7 14)(8 13)(11 16)(12 15)

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

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

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

G:=TransitiveGroup(16,163);

C22.SD16 is a maximal subgroup of
C42.375D4  C24.53D4  C24.150D4  C42.55D4  C24.54D4  C42.57D4  C24.56D4  C42.58D4  C24.58D4  C42.59D4  C24.59D4  C42.63D4  C23⋊D8  C4⋊C4.D4  (C2×C4)⋊D8  C24.9D4  C232SD16  C4⋊C4.6D4  Q8⋊D4⋊C2  C24.12D4  C4⋊C4.12D4  C24.15D4  C24.16D4  C4⋊C4.18D4  C4⋊C4.19D4  C24.18D4  C23.8S4  D10.1D8  D10.SD16
 C23.D4p: C24.60D4  C23.5D8  C22.2D24  C22.2D40  C22.2D56 ...
 (C2×C2p).D8: C42.403D4  C42.61D4  (C2×C4).5D8  C6.C4≀C2  (C6×D4)⋊C4  (C2×D20)⋊C4  C4⋊C4⋊Dic5  C14.C4≀C2 ...
C22.SD16 is a maximal quotient of
(C2×C4).98D8  C4⋊C4⋊C8  C23.30D8  C24.D4  C23.4D8  C2.C2≀C4  (C2×C4).D8
 C22.D8p: C22.SD32  C22.2D24  C22.2D40  C22.2D56 ...
 C2p.C4≀C2: C23.D8  C23.2D8  C23.SD16  C23.2SD16  C23.32D8  C23.12SD16  C23.13SD16  C6.C4≀C2 ...

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

16000
01600
00160
0091
,
1000
0100
00160
00016
,
14300
141400
0028
00015
,
141400
14300
0092
00118
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,9,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,2,0,0,0,8,15],[14,14,0,0,14,3,0,0,0,0,9,11,0,0,2,8] >;

C22.SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(64,8);
// by ID

G=gap.SmallGroup(64,8);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681]);
// Polycyclic

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

Export

Subgroup lattice of C22.SD16 in TeX
Character table of C22.SD16 in TeX

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