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## G = Dic3⋊5D12order 288 = 25·32

### 2nd semidirect product of Dic3 and D12 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3⋊5D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — Dic3⋊5D12
 Lower central C32 — C3×C6 — Dic3⋊5D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic35D12
G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 962 in 215 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×2], Dic3 [×3], C12 [×4], C12 [×7], D6 [×24], C2×C6 [×2], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×8], D12 [×12], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×5], C22×S3 [×6], C4×D4, C3×Dic3 [×2], C3×Dic3 [×3], C3×C12 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C4×Dic3, C4⋊Dic3, D6⋊C4 [×4], C4×C12, C3×C4⋊C4, S3×C2×C4 [×4], C2×D12 [×3], C6.D6 [×4], C6×Dic3 [×2], C6×Dic3 [×2], C12⋊S3 [×4], C6×C12, C22×C3⋊S3 [×2], C4×D12, Dic35D4, C6.D12 [×2], Dic3×C12, C3×C4⋊Dic3, C2×C6.D6 [×2], C2×C12⋊S3, Dic35D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×4], D12 [×2], C22×S3 [×2], C4×D4, S32, S3×C2×C4 [×2], C2×D12, C4○D12, S3×D4, Q83S3, C6.D6 [×2], C2×S32, C4×D12, Dic35D4, D6.6D6, S3×D12, C2×C6.D6, Dic35D12

Smallest permutation representation of Dic35D12
On 48 points
Generators in S48
(1 34 9 30 5 26)(2 35 10 31 6 27)(3 36 11 32 7 28)(4 25 12 33 8 29)(13 44 17 48 21 40)(14 45 18 37 22 41)(15 46 19 38 23 42)(16 47 20 39 24 43)
(1 45 30 22)(2 46 31 23)(3 47 32 24)(4 48 33 13)(5 37 34 14)(6 38 35 15)(7 39 36 16)(8 40 25 17)(9 41 26 18)(10 42 27 19)(11 43 28 20)(12 44 29 21)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(38 48)(39 47)(40 46)(41 45)(42 44)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 18 18 18 18 2 2 4 2 2 3 3 3 3 6 ··· 6 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 S3 D4 D6 D6 C4○D4 D12 C4×S3 C4○D12 S32 S3×D4 Q8⋊3S3 C6.D6 C2×S32 D6.6D6 S3×D12 kernel Dic3⋊5D12 C6.D12 Dic3×C12 C3×C4⋊Dic3 C2×C6.D6 C2×C12⋊S3 C12⋊S3 C4×Dic3 C4⋊Dic3 C3×Dic3 C2×Dic3 C2×C12 C3×C6 Dic3 C12 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 1 2 4 2 2 4 8 4 1 1 1 2 1 2 2

Matrix representation of Dic35D12 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 3 0 0 0 0 3 4 0 0 0 0 0 0 0 12 0 0 0 0 1 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 7 12 0 0 0 0 0 0 12 12 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;

Dic35D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_5D_{12}
% in TeX

G:=Group("Dic3:5D12");
// GroupNames label

G:=SmallGroup(288,542);
// by ID

G=gap.SmallGroup(288,542);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,219,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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