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## G = SD16⋊C4order 64 = 26

### 1st semidirect product of SD16 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — SD16⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — SD16⋊C4
 Lower central C1 — C2 — C4 — SD16⋊C4
 Upper central C1 — C22 — C42 — SD16⋊C4
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊C4

Generators and relations for SD16⋊C4
G = < a,b,c | a8=b2=c4=1, bab=a3, cac-1=a5, bc=cb >

Subgroups: 101 in 60 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, SD16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22, C8.C22, SD16⋊C4

Character table of SD16⋊C4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D size 1 1 1 1 4 4 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 1 -1 1 -1 1 -1 1 -1 -i i -i i -i -i i -1 1 i i -1 -i 1 linear of order 4 ρ10 1 -1 1 -1 -1 1 1 -1 i -i i -i -i -i i 1 -1 i -i -1 i 1 linear of order 4 ρ11 1 -1 1 -1 -1 1 1 -1 -i i -i i -i i i -1 1 -i -i 1 i -1 linear of order 4 ρ12 1 -1 1 -1 1 -1 1 -1 i -i i -i -i i i 1 -1 -i i 1 -i -1 linear of order 4 ρ13 1 -1 1 -1 -1 1 1 -1 i -i i -i i -i -i -1 1 i i 1 -i -1 linear of order 4 ρ14 1 -1 1 -1 1 -1 1 -1 -i i -i i i -i -i 1 -1 i -i 1 i -1 linear of order 4 ρ15 1 -1 1 -1 1 -1 1 -1 i -i i -i i i -i -1 1 -i -i -1 i 1 linear of order 4 ρ16 1 -1 1 -1 -1 1 1 -1 -i i -i i i i -i 1 -1 -i i -1 -i 1 linear of order 4 ρ17 2 2 2 2 0 0 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 0 0 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 0 0 -2 2 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 0 -2 2 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 4 -4 -4 0 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

Smallest permutation representation of SD16⋊C4
On 32 points
Generators in S32
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 32)(2 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)
(1 11 32 19)(2 16 25 24)(3 13 26 21)(4 10 27 18)(5 15 28 23)(6 12 29 20)(7 9 30 17)(8 14 31 22)

G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18), (1,11,32,19)(2,16,25,24)(3,13,26,21)(4,10,27,18)(5,15,28,23)(6,12,29,20)(7,9,30,17)(8,14,31,22)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18), (1,11,32,19)(2,16,25,24)(3,13,26,21)(4,10,27,18)(5,15,28,23)(6,12,29,20)(7,9,30,17)(8,14,31,22) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,32),(2,27),(3,30),(4,25),(5,28),(6,31),(7,26),(8,29),(9,21),(10,24),(11,19),(12,22),(13,17),(14,20),(15,23),(16,18)], [(1,11,32,19),(2,16,25,24),(3,13,26,21),(4,10,27,18),(5,15,28,23),(6,12,29,20),(7,9,30,17),(8,14,31,22)]])

Matrix representation of SD16⋊C4 in GL6(𝔽17)

 0 8 0 0 0 0 2 0 0 0 0 0 0 0 0 8 0 11 0 0 13 8 3 11 0 0 0 11 0 9 0 0 3 11 4 9
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 16 1 0 0 0 0 0 0 16 0 0 0 0 0 16 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 0 0

G:=sub<GL(6,GF(17))| [0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,13,0,3,0,0,8,8,11,11,0,0,0,3,0,4,0,0,11,11,9,9],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

SD16⋊C4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes C_4
% in TeX

G:=Group("SD16:C4");
// GroupNames label

G:=SmallGroup(64,121);
// by ID

G=gap.SmallGroup(64,121);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^3,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations

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