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## G = D6⋊6SD16order 192 = 26·3

### 2nd semidirect product of D6 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D6⋊6SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×S3×D4 — D6⋊6SD16
 Lower central C3 — C6 — C2×C12 — D6⋊6SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for D66SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c3 >

Subgroups: 696 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, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, Q82S3, C2×C24, C3×SD16, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22⋊SD16, D6⋊C8, C2.D24, D4⋊Dic3, C2×Q82S3, D63Q8, C6×SD16, C2×S3×D4, D66SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×SD16, C8⋊C22, S3×D4, C2×C3⋊D4, C22⋊SD16, S3×SD16, Q83D6, C232D6, D66SD16

Smallest permutation representation of D66SD16
On 48 points
Generators in S48
(1 24 47 28 13 33)(2 17 48 29 14 34)(3 18 41 30 15 35)(4 19 42 31 16 36)(5 20 43 32 9 37)(6 21 44 25 10 38)(7 22 45 26 11 39)(8 23 46 27 12 40)
(1 33)(2 48)(3 35)(4 42)(5 37)(6 44)(7 39)(8 46)(9 20)(11 22)(13 24)(15 18)(25 38)(26 45)(27 40)(28 47)(29 34)(30 41)(31 36)(32 43)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(34 36)(35 39)(38 40)(41 45)(42 48)(44 46)

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

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

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

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 C3⋊D4 C8⋊C22 S3×D4 S3×D4 S3×SD16 Q8⋊3D6 kernel D6⋊6SD16 D6⋊C8 C2.D24 D4⋊Dic3 C2×Q8⋊2S3 D6⋊3Q8 C6×SD16 C2×S3×D4 C2×SD16 D12 C2×Dic3 C3×D4 C22×S3 C2×C8 C2×D4 C2×Q8 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 D66SD16 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 0 0 0 72
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 67 67 0 0 0 0 6 67 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

D66SD16 in GAP, Magma, Sage, TeX

D_6\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("D6:6SD16");
// GroupNames label

G:=SmallGroup(192,728);
// by ID

G=gap.SmallGroup(192,728);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,851,438,102,6278]);
// Polycyclic

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

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