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## G = D4×SD16order 128 = 27

### Direct product of D4 and SD16

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4×SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 624 in 273 conjugacy classes, 104 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, SD16, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C41D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C2×SD16, C22×D4, C22×D4, C22×Q8, C8×D4, C4×SD16, Q8⋊D4, C22⋊SD16, C4⋊SD16, D4.D4, C88D4, C85D4, D42, D4×Q8, C22×SD16, D4×SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C2×SD16, C22×D4, 2+ 1+4, D42, C22×SD16, D4○SD16, D4×SD16

Smallest permutation representation of D4×SD16
On 32 points
Generators in S32
(1 14 20 32)(2 15 21 25)(3 16 22 26)(4 9 23 27)(5 10 24 28)(6 11 17 29)(7 12 18 30)(8 13 19 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)

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

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

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

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 D4 SD16 2+ 1+4 D4○SD16 kernel D4×SD16 C8×D4 C4×SD16 Q8⋊D4 C22⋊SD16 C4⋊SD16 D4.D4 C8⋊8D4 C8⋊5D4 D42 D4×Q8 C22×SD16 C22⋊C4 C4⋊C4 SD16 C2×D4 D4 C4 C2 # reps 1 1 1 2 2 1 1 2 1 1 1 2 2 1 4 1 8 1 2

Matrix representation of D4×SD16 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 1 15 0 0 1 16
,
 16 0 0 0 0 16 0 0 0 0 1 15 0 0 0 16
,
 10 10 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 16 16 0 0 0 0 16 0 0 0 0 16
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,15,16],[10,12,0,0,10,0,0,0,0,0,1,0,0,0,0,1],[1,16,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;

D4×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("D4xSD16");
// GroupNames label

G:=SmallGroup(128,2013);
// by ID

G=gap.SmallGroup(128,2013);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,346,2804,1411,375,172]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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