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## G = SD16⋊D4order 128 = 27

### 1st semidirect product of SD16 and D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for SD16⋊D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 648 in 273 conjugacy classes, 96 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C89D4, SD16⋊C4, C22⋊D8, Q8⋊D4, D4⋊D4, C22⋊SD16, C4⋊SD16, D4.2D4, C87D4, C8⋊D4, C83D4, D42, Q85D4, C22×SD16, C2×C8⋊C22, SD16⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2+ 1+4, D42, C2×C8⋊C22, D4○SD16, SD16⋊D4

Character table of SD16⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F size 1 1 1 1 2 2 4 4 4 8 8 8 2 2 4 4 4 4 4 4 8 8 8 4 4 4 4 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 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 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 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 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 -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 -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 -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 -1 1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 -1 -1 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 ρ10 1 1 1 1 -1 -1 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 ρ11 1 1 1 1 1 1 -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 ρ12 1 1 1 1 1 1 -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 ρ13 1 1 1 1 -1 -1 -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 ρ14 1 1 1 1 -1 -1 -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 ρ15 1 1 1 1 1 1 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 ρ16 1 1 1 1 1 1 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 ρ17 2 -2 2 -2 0 0 2 -2 0 0 0 0 -2 2 -2 0 2 0 0 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 2 0 0 2 0 0 0 -2 -2 0 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 0 0 -2 0 0 0 -2 -2 0 -2 0 2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 0 0 -2 2 0 0 0 0 -2 2 2 0 -2 0 0 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 0 0 2 -2 0 0 0 0 -2 2 2 0 -2 0 0 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 2 -2 -2 0 0 -2 0 0 0 -2 -2 0 2 0 2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 2 -2 -2 0 0 2 0 0 0 -2 -2 0 2 0 -2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 -2 2 -2 0 0 -2 2 0 0 0 0 -2 2 -2 0 2 0 0 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ25 4 -4 -4 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ27 4 -4 -4 4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 0 0 2√-2 0 0 complex lifted from D4○SD16 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 0 0 -2√-2 0 0 complex lifted from D4○SD16

Smallest permutation representation of SD16⋊D4
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 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 26 23 12)(2 25 24 11)(3 32 17 10)(4 31 18 9)(5 30 19 16)(6 29 20 15)(7 28 21 14)(8 27 22 13)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

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

Matrix representation of SD16⋊D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 16 16 1 2 0 0 0 1 0 0 0 0 16 16 1 1
,
 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 16 16 1 2 0 0 1 0 0 16
,
 2 2 0 0 0 0 6 15 0 0 0 0 0 0 16 5 7 7 0 0 12 1 0 10 0 0 6 6 6 12 0 0 0 11 12 11
,
 2 2 0 0 0 0 7 15 0 0 0 0 0 0 16 5 7 7 0 0 12 1 0 10 0 0 6 6 6 12 0 0 0 11 12 11

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,16,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[2,6,0,0,0,0,2,15,0,0,0,0,0,0,16,12,6,0,0,0,5,1,6,11,0,0,7,0,6,12,0,0,7,10,12,11],[2,7,0,0,0,0,2,15,0,0,0,0,0,0,16,12,6,0,0,0,5,1,6,11,0,0,7,0,6,12,0,0,7,10,12,11] >;

SD16⋊D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_4
% in TeX

G:=Group("SD16:D4");
// GroupNames label

G:=SmallGroup(128,1997);
// by ID

G=gap.SmallGroup(128,1997);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,346,2804,1411,375,172]);
// Polycyclic

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

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