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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C22.SD16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C4⋊D4 — C22.SD16
 Lower central C1 — C22 — C2×C4 — C22.SD16
 Upper central C1 — C22 — C22×C4 — C22.SD16
 Jennings C1 — C2 — C22 — C22×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 >

Character table of C22.SD16

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D size 1 1 1 1 2 2 8 2 2 4 4 4 4 4 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 -i i 1 i -i 1 i i -i -i linear of order 4 ρ6 1 1 1 1 -1 -1 1 -1 -1 -i i 1 i -i -1 -i -i i i linear of order 4 ρ7 1 1 1 1 -1 -1 1 -1 -1 i -i 1 -i i -1 i i -i -i linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 -1 i -i 1 -i i 1 -i -i i i linear of order 4 ρ9 2 2 2 2 2 2 0 -2 -2 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 2 2 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ12 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ13 2 -2 2 -2 0 0 0 -2i 2i -1+i -1-i 0 1+i 1-i 0 0 0 0 0 complex lifted from C4≀C2 ρ14 2 -2 2 -2 0 0 0 2i -2i 1+i 1-i 0 -1+i -1-i 0 0 0 0 0 complex lifted from C4≀C2 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 0 0 0 2i -2i -1-i -1+i 0 1-i 1+i 0 0 0 0 0 complex lifted from C4≀C2 ρ17 2 -2 2 -2 0 0 0 -2i 2i 1-i 1+i 0 -1-i -1+i 0 0 0 0 0 complex lifted from C4≀C2 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 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);

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

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 9 1
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 14 3 0 0 14 14 0 0 0 0 2 8 0 0 0 15
,
 14 14 0 0 14 3 0 0 0 0 9 2 0 0 11 8
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

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