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## G = SD16⋊10D4order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — SD16⋊10D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C22×D4 — C22×SD16 — SD16⋊10D4
 Lower central C1 — C2 — C2×C4 — SD16⋊10D4
 Upper central C1 — C22 — C4×D4 — SD16⋊10D4
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊10D4

Generators and relations for SD1610D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, ad=da, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 544 in 252 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, D8, SD16, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C22⋊C4, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C22×C8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8, C22×D4, C22×Q8, C2×C4○D4, C8×D4, C4×SD16, C22⋊D8, D4⋊D4, C22⋊Q16, D4.7D4, D4.2D4, Q8.D4, C88D4, C8.12D4, D45D4, Q85D4, C22×SD16, C2×C4○D8, SD1610D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2+ 1+4, D42, C2×C4○D8, D4○SD16, SD1610D4

Smallest permutation representation of SD1610D4
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 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)
(1 16 20 26)(2 9 21 27)(3 10 22 28)(4 11 23 29)(5 12 24 30)(6 13 17 31)(7 14 18 32)(8 15 19 25)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D 8E ··· 8J order 1 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 4 4 4 8 8 2 ··· 2 4 4 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 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D8 2+ 1+4 D4○SD16 kernel SD16⋊10D4 C8×D4 C4×SD16 C22⋊D8 D4⋊D4 C22⋊Q16 D4.7D4 D4.2D4 Q8.D4 C8⋊8D4 C8.12D4 D4⋊5D4 Q8⋊5D4 C22×SD16 C2×C4○D8 C22⋊C4 C4⋊C4 SD16 C2×D4 C22 C4 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 4 1 8 1 2

Matrix representation of SD1610D4 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 7 7 0 0 5 0
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 16 16
,
 16 1 0 0 15 1 0 0 0 0 4 8 0 0 13 13
,
 16 1 0 0 0 1 0 0 0 0 4 8 0 0 13 13
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,7,5,0,0,7,0],[16,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[16,15,0,0,1,1,0,0,0,0,4,13,0,0,8,13],[16,0,0,0,1,1,0,0,0,0,4,13,0,0,8,13] >;

SD1610D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_{10}D_4
% in TeX

G:=Group("SD16:10D4");
// GroupNames label

G:=SmallGroup(128,2014);
// by ID

G=gap.SmallGroup(128,2014);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2019,346,2804,1411,375,172]);
// Polycyclic

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

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