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## G = C23⋊SD16order 128 = 27

### 1st semidirect product of C23 and SD16 acting via SD16/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C23⋊SD16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23⋊2Q8 — C23⋊SD16
 Lower central C1 — C22 — C22×C4 — C23⋊SD16
 Upper central C1 — C22 — C22×C4 — C23⋊SD16
 Jennings C1 — C2 — C22 — C22×C4 — C23⋊SD16

Generators and relations for C23⋊SD16
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 460 in 158 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C24, C2.C42, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22⋊Q8, C22⋊Q8, C2×SD16, C22×D4, C22×D4, C23⋊C8, C23.31D4, C232D4, C22⋊SD16, C232Q8, C23⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8.C22, Q8⋊D4, D44D4, C2≀C22, C23⋊SD16

Character table of C23⋊SD16

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 8 8 4 4 8 8 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 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 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 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 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 -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 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 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 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 2 2 -2 -2 0 0 -2 -2 0 0 0 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 0 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 0 0 -2 2 0 0 -2 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 0 0 2 -2 0 2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 2 2 2 0 0 -2 -2 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 0 0 0 0 -2 2 0 0 2 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ16 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ17 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ18 2 2 -2 -2 -2 2 -2 2 0 0 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 -2 0 0 0 0 0 2 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ20 4 -4 4 -4 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ21 4 -4 -4 4 0 0 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 0 0 0 orthogonal lifted from D4⋊4D4 ρ22 4 -4 4 -4 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ23 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Permutation representations of C23⋊SD16
On 16 points - transitive group 16T405
Generators in S16
(2 10)(3 11)(6 14)(7 15)
(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)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)

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

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

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

G:=TransitiveGroup(16,405);

Matrix representation of C23⋊SD16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 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 0 0 0 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 7 5 0 0 0 0 7 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 1 0 0 0
,
 16 0 0 0 0 0 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[7,7,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[16,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C23⋊SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,328);
// by ID

G=gap.SmallGroup(128,328);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,232,422,1123,570,521,136,1411]);
// Polycyclic

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

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