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## G = D4⋊7SD16order 128 = 27

### 1st semidirect product of D4 and SD16 acting through Inn(D4)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4⋊7SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — D42 — D4⋊7SD16
 Lower central C1 — C2 — C2×C4 — D4⋊7SD16
 Upper central C1 — C22 — C4×D4 — D4⋊7SD16
 Jennings C1 — C2 — C2 — C2×C4 — D4⋊7SD16

Generators and relations for D47SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 552 in 231 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C42.C2, C41D4, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×D4, C2×D4⋊C4, C8×D4, C4×SD16, C22⋊SD16, C4⋊SD16, C88D4, D42Q8, C23.46D4, C4.4D8, D42, D43Q8, D47SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C22×SD16, D4○D8, D47SD16

Smallest permutation representation of D47SD16
On 32 points
Generators in S32
(1 9 30 19)(2 10 31 20)(3 11 32 21)(4 12 25 22)(5 13 26 23)(6 14 27 24)(7 15 28 17)(8 16 29 18)
(1 23)(2 14)(3 17)(4 16)(5 19)(6 10)(7 21)(8 12)(9 26)(11 28)(13 30)(15 32)(18 25)(20 27)(22 29)(24 31)
(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 26)(2 29)(3 32)(4 27)(5 30)(6 25)(7 28)(8 31)(9 23)(10 18)(11 21)(12 24)(13 19)(14 22)(15 17)(16 20)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4I 4J 4K 4L 4M 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 2 2 2 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 SD16 C4○D4 2+ 1+4 D4○D8 kernel D4⋊7SD16 C2×D4⋊C4 C8×D4 C4×SD16 C22⋊SD16 C4⋊SD16 C8⋊8D4 D4⋊2Q8 C23.46D4 C4.4D8 D42 D4⋊3Q8 C22⋊C4 C4⋊C4 C2×D4 D4 D4 C4 C2 # reps 1 2 1 1 2 1 2 1 2 1 1 1 2 1 1 8 4 1 2

Matrix representation of D47SD16 in GL4(𝔽17) generated by

 1 2 0 0 16 16 0 0 0 0 16 0 0 0 0 16
,
 1 2 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 13 9 0 0 4 4 0 0 0 0 5 12 0 0 5 5
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1
G:=sub<GL(4,GF(17))| [1,16,0,0,2,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,2,16,0,0,0,0,1,0,0,0,0,1],[13,4,0,0,9,4,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

D47SD16 in GAP, Magma, Sage, TeX

D_4\rtimes_7{\rm SD}_{16}
% in TeX

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

G:=SmallGroup(128,2027);
// by ID

G=gap.SmallGroup(128,2027);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,346,4037,1027,124]);
// Polycyclic

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

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