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## G = D6⋊5SD16order 192 = 26·3

### 1st semidirect product of D6 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D6⋊5SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×S3×D4 — D6⋊5SD16
 Lower central C3 — C6 — C2×C12 — D6⋊5SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D65SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd=c3 >

Subgroups: 712 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C24⋊C2, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4.S3, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C22⋊SD16, C6.D8, D6⋊C8, C3×D4⋊C4, C4.D12, C2×C24⋊C2, C2×D4.S3, C2×S3×D4, D65SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C22≀C2, C2×SD16, C8⋊C22, C2×D12, S3×D4, C22⋊SD16, D6⋊D4, D8⋊S3, S3×SD16, D65SD16

Smallest permutation representation of D65SD16
On 48 points
Generators in S48
(1 40 26 22 42 12)(2 13 43 23 27 33)(3 34 28 24 44 14)(4 15 45 17 29 35)(5 36 30 18 46 16)(6 9 47 19 31 37)(7 38 32 20 48 10)(8 11 41 21 25 39)
(1 16)(2 47)(3 10)(4 41)(5 12)(6 43)(7 14)(8 45)(9 13)(11 15)(17 39)(18 26)(19 33)(20 28)(21 35)(22 30)(23 37)(24 32)(25 29)(27 31)(34 48)(36 42)(38 44)(40 46)
(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 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)(33 45)(34 48)(35 43)(36 46)(37 41)(38 44)(39 47)(40 42)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 D6 SD16 D12 C8⋊C22 S3×D4 S3×D4 D8⋊S3 S3×SD16 kernel D6⋊5SD16 C6.D8 D6⋊C8 C3×D4⋊C4 C4.D12 C2×C24⋊C2 C2×D4.S3 C2×S3×D4 D4⋊C4 D12 C2×Dic3 C3×D4 C22×S3 C4⋊C4 C2×C8 C2×D4 D6 D4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 4 4 1 1 1 2 2

Matrix representation of D65SD16 in GL4(𝔽73) generated by

 0 1 0 0 72 1 0 0 0 0 1 0 0 0 0 1
,
 72 1 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 59 7 0 0 66 14 0 0 0 0 67 67 0 0 6 67
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[59,66,0,0,7,14,0,0,0,0,67,6,0,0,67,67],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;

D65SD16 in GAP, Magma, Sage, TeX

D_6\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("D6:5SD16");
// GroupNames label

G:=SmallGroup(192,335);
// by ID

G=gap.SmallGroup(192,335);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,135,268,570,297,136,6278]);
// Polycyclic

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

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