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## G = SD16⋊13D6order 192 = 26·3

### 2nd semidirect product of SD16 and D6 acting through Inn(SD16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — SD16⋊13D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — D4⋊6D6 — SD16⋊13D6
 Lower central C3 — C6 — C12 — SD16⋊13D6
 Upper central C1 — C2 — C2×C4 — C2×SD16

Generators and relations for SD1613D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a3, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 712 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C8○D4, C2×SD16, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C3×SD16, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, S3×Q8, Q83S3, Q83S3, C2×C3⋊D4, C6×D4, C6×Q8, D4○SD16, C8○D12, C4○D24, S3×SD16, Q83D6, D4.D6, Q8.7D6, D126C22, Q8.11D6, C6×SD16, D46D6, Q8.15D6, SD1613D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○SD16, C2×S3×D4, SD1613D6

Smallest permutation representation of SD1613D6
On 48 points
Generators in S48
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 42)(10 45)(11 48)(12 43)(13 46)(14 41)(15 44)(16 47)(25 35)(26 38)(27 33)(28 36)(29 39)(30 34)(31 37)(32 40)
(1 43 37)(2 46 38 4 44 40)(3 41 39 7 45 35)(5 47 33)(6 42 34 8 48 36)(9 30 17 11 28 19)(10 25 18 14 29 22)(12 31 20)(13 26 21 15 32 23)(16 27 24)
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 13)(10 14)(11 15)(12 16)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)

G:=sub<Sym(48)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(25,35)(26,38)(27,33)(28,36)(29,39)(30,34)(31,37)(32,40), (1,43,37)(2,46,38,4,44,40)(3,41,39,7,45,35)(5,47,33)(6,42,34,8,48,36)(9,30,17,11,28,19)(10,25,18,14,29,22)(12,31,20)(13,26,21,15,32,23)(16,27,24), (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,13)(10,14)(11,15)(12,16)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(25,35)(26,38)(27,33)(28,36)(29,39)(30,34)(31,37)(32,40), (1,43,37)(2,46,38,4,44,40)(3,41,39,7,45,35)(5,47,33)(6,42,34,8,48,36)(9,30,17,11,28,19)(10,25,18,14,29,22)(12,31,20)(13,26,21,15,32,23)(16,27,24), (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,13)(10,14)(11,15)(12,16)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,20),(2,23),(3,18),(4,21),(5,24),(6,19),(7,22),(8,17),(9,42),(10,45),(11,48),(12,43),(13,46),(14,41),(15,44),(16,47),(25,35),(26,38),(27,33),(28,36),(29,39),(30,34),(31,37),(32,40)], [(1,43,37),(2,46,38,4,44,40),(3,41,39,7,45,35),(5,47,33),(6,42,34,8,48,36),(9,30,17,11,28,19),(10,25,18,14,29,22),(12,31,20),(13,26,21,15,32,23),(16,27,24)], [(1,37),(2,38),(3,39),(4,40),(5,33),(6,34),(7,35),(8,36),(9,13),(10,14),(11,15),(12,16),(17,32),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31)]])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 2 4 4 6 6 12 12 2 2 2 4 4 6 6 12 12 2 2 2 8 8 2 2 4 12 12 4 4 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 S3×D4 S3×D4 D4○SD16 SD16⋊13D6 kernel SD16⋊13D6 C8○D12 C4○D24 S3×SD16 Q8⋊3D6 D4.D6 Q8.7D6 D12⋊6C22 Q8.11D6 C6×SD16 D4⋊6D6 Q8.15D6 C2×SD16 Dic6 D12 C3⋊D4 C2×C8 SD16 C2×D4 C2×Q8 C4 C22 C3 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 1 4 1 1 1 1 2 4

Matrix representation of SD1613D6 in GL8(𝔽73)

 0 0 71 0 0 0 0 0 0 0 0 71 0 0 0 0 37 0 0 0 0 0 0 0 0 37 0 0 0 0 0 0 0 0 0 0 61 12 0 0 0 0 0 0 67 0 0 0 0 0 0 0 43 60 0 69 0 0 0 0 56 17 18 61
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 39 34 36 61 0 0 0 0 56 17 18 0 0 0 0 0 61 8 17 13 0 0 0 0 6 61 0 0
,
 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 68 10 1 48 0 0 0 0 0 0 0 72
,
 72 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 10 63 72 0 0 0 0 0 0 0 0 72

G:=sub<GL(8,GF(73))| [0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,71,0,0,0,0,0,0,0,0,71,0,0,0,0,0,0,0,0,0,0,61,67,43,56,0,0,0,0,12,0,60,17,0,0,0,0,0,0,0,18,0,0,0,0,0,0,69,61],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,56,61,6,0,0,0,0,34,17,8,61,0,0,0,0,36,18,17,0,0,0,0,0,61,0,13,0],[72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,68,0,0,0,0,0,0,72,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,48,72],[72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,63,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72] >;

SD1613D6 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_{13}D_6
% in TeX

G:=Group("SD16:13D6");
// GroupNames label

G:=SmallGroup(192,1321);
// by ID

G=gap.SmallGroup(192,1321);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,185,136,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^3,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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