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## G = SD16⋊7D4order 128 = 27

### 3rd 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⋊7D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C2×C4○D4 — C2×C4○D8 — SD16⋊7D4
 Lower central C1 — C2 — C2×C4 — SD16⋊7D4
 Upper central C1 — C22 — C4×D4 — SD16⋊7D4
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊7D4

Generators and relations for SD167D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a-1, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 584 in 255 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, 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, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C22⋊C4, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C41D4, C41D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C89D4, SD16⋊C4, C22⋊D8, D4⋊D4, D4.7D4, C4⋊D8, Q8.D4, C87D4, C8⋊D4, C83D4, D45D4, Q86D4, C2×C4○D8, C2×C8⋊C22, SD167D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8⋊C22, D4○D8, SD167D4

Character table of SD167D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 4 8 8 8 2 2 2 2 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 0 -2 0 0 0 2 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 0 0 -2 2 0 0 0 0 2 -2 0 2 0 0 0 -2 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 2 0 0 0 0 0 2 -2 -2 2 0 -2 -2 2 0 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 0 -2 0 0 0 2 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 0 0 2 -2 0 0 0 0 2 -2 0 2 0 0 0 -2 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ22 2 2 2 2 2 -2 0 0 0 0 0 -2 -2 -2 -2 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 0 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 2 -2 -2 0 0 0 0 0 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 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 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 orthogonal lifted from D4○D8 ρ27 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 orthogonal lifted from D4○D8 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 -4i 0 0 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22 ρ29 4 -4 -4 4 0 0 0 0 0 0 0 4i 0 0 -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22

Smallest permutation representation of SD167D4
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 16)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)
(1 24 12 31)(2 23 13 30)(3 22 14 29)(4 21 15 28)(5 20 16 27)(6 19 9 26)(7 18 10 25)(8 17 11 32)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)

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

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

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

Matrix representation of SD167D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 14 3 0 0 0 0 14 14 0 0
,
 1 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 0 0 0 0 0 0 1
,
 16 15 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 16 0 0 0 0 0 1 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 1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,3,3,0,0,0,0,14,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

SD167D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_7D_4
% in TeX

G:=Group("SD16:7D4");
// GroupNames label

G:=SmallGroup(128,2000);
// by ID

G=gap.SmallGroup(128,2000);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,346,2804,1411,375,172]);
// 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,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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