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## G = C23.SD16order 128 = 27

### 1st non-split extension by C23 of SD16 acting via SD16/C2=D4

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C23.SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C8⋊2D4 — C23.SD16
 Lower central C1 — C2 — C2×C4 — C2×C8 — C23.SD16
 Upper central C1 — C2 — C2×C4 — C2×M4(2) — C23.SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C23.SD16

Generators and relations for C23.SD16
G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, dad-1=eae=ab=ba, ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=abd3 >

Character table of C23.SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 16A 16B 16C 16D size 1 1 2 4 16 2 2 4 16 4 4 8 8 8 8 8 8 8 8 8 ρ1 1 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 -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 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 -1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 -1 -1 -1 -i i -i 1 i i -i i -i linear of order 4 ρ6 1 1 1 -1 -1 1 1 -1 1 -1 -1 i -i i 1 -i i -i i -i linear of order 4 ρ7 1 1 1 -1 -1 1 1 -1 1 -1 -1 -i i -i 1 i -i i -i i linear of order 4 ρ8 1 1 1 -1 1 1 1 -1 -1 -1 -1 i -i i 1 -i -i i -i i linear of order 4 ρ9 2 2 2 2 0 2 2 2 0 -2 -2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 2 2 -2 0 2 2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ12 2 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 -2 0 0 2 -2 0 0 -2i 2i 1+i 1-i -1-i 0 -1+i 0 0 0 0 complex lifted from C4≀C2 ρ14 2 2 -2 0 0 2 -2 0 0 -2i 2i -1-i -1+i 1+i 0 1-i 0 0 0 0 complex lifted from C4≀C2 ρ15 2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ16 2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 2 -2 0 0 2 -2 0 0 2i -2i -1+i -1-i 1-i 0 1+i 0 0 0 0 complex lifted from C4≀C2 ρ18 2 2 -2 0 0 2 -2 0 0 2i -2i 1-i 1+i -1+i 0 -1-i 0 0 0 0 complex lifted from C4≀C2 ρ19 4 4 -4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C23.SD16
On 16 points - transitive group 16T348
Generators in S16
(1 9)(2 10)(5 13)(6 14)
(2 10)(4 12)(6 14)(8 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 12)(3 7)(4 10)(5 13)(6 8)(11 15)(14 16)

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

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

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

G:=TransitiveGroup(16,348);

Matrix representation of C23.SD16 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C23.SD16 in GAP, Magma, Sage, TeX

C_2^3.{\rm SD}_{16}
% in TeX

G:=Group("C2^3.SD16");
// GroupNames label

G:=SmallGroup(128,73);
// by ID

G=gap.SmallGroup(128,73);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,521,248,1411,172,4037,2028,124]);
// Polycyclic

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

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