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## G = Dic3⋊4D12order 288 = 25·32

### 1st semidirect product of Dic3 and D12 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3⋊4D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — Dic3⋊4D12
 Lower central C32 — C3×C6 — Dic3⋊4D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic34D12
G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 826 in 205 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×7], C22, C22 [×8], S3 [×8], C6 [×6], C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×4], Dic3 [×4], C12 [×8], D6 [×2], D6 [×14], C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×6], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×3], C22×C6, C4×D4, C3×Dic3 [×4], C3×Dic3, C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4 [×3], C4×C12, C3×C22⋊C4, S3×C2×C4 [×3], C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3 [×2], C6.D6 [×2], C3⋊D12 [×4], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C4×D12, Dic34D4, Dic3⋊Dic3, Dic3×C12, C3×D6⋊C4, C6.11D12, C2×S3×Dic3, C2×C6.D6, C2×C3⋊D12, Dic34D12
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, D42S3, C2×S32, C4×D12, Dic34D4, C4×S32, S3×D12, D6.3D6, Dic34D12

Smallest permutation representation of Dic34D12
On 48 points
Generators in S48
(1 40 9 48 5 44)(2 45 6 37 10 41)(3 42 11 38 7 46)(4 47 8 39 12 43)(13 25 21 33 17 29)(14 30 18 34 22 26)(15 27 23 35 19 31)(16 32 20 36 24 28)
(1 14 48 34)(2 15 37 35)(3 16 38 36)(4 17 39 25)(5 18 40 26)(6 19 41 27)(7 20 42 28)(8 21 43 29)(9 22 44 30)(10 23 45 31)(11 24 46 32)(12 13 47 33)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 17)(14 16)(18 24)(19 23)(20 22)(25 33)(26 32)(27 31)(28 30)(34 36)(38 48)(39 47)(40 46)(41 45)(42 44)

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

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

G=PermutationGroup([(1,40,9,48,5,44),(2,45,6,37,10,41),(3,42,11,38,7,46),(4,47,8,39,12,43),(13,25,21,33,17,29),(14,30,18,34,22,26),(15,27,23,35,19,31),(16,32,20,36,24,28)], [(1,14,48,34),(2,15,37,35),(3,16,38,36),(4,17,39,25),(5,18,40,26),(6,19,41,27),(7,20,42,28),(8,21,43,29),(9,22,44,30),(10,23,45,31),(11,24,46,32),(12,13,47,33)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,17),(14,16),(18,24),(19,23),(20,22),(25,33),(26,32),(27,31),(28,30),(34,36),(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 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 size 1 1 1 1 6 6 18 18 2 2 4 2 2 3 3 3 3 6 6 6 6 18 18 2 ··· 2 4 4 4 12 12 2 2 2 2 4 ··· 4 6 ··· 6 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D4 D6 D6 D6 C4○D4 C4×S3 D12 C4×S3 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 C4×S32 S3×D12 D6.3D6 kernel Dic3⋊4D12 Dic3⋊Dic3 Dic3×C12 C3×D6⋊C4 C6.11D12 C2×S3×Dic3 C2×C6.D6 C2×C3⋊D12 C3⋊D12 C4×Dic3 D6⋊C4 C3×Dic3 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 Dic3 D6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 3 2 1 2 4 4 4 4 1 1 1 1 2 2 2

Matrix representation of Dic34D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 12 0
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 12 3 0 0 0 0 8 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 8 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

Dic34D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_4D_{12}
% in TeX

G:=Group("Dic3:4D12");
// GroupNames label

G:=SmallGroup(288,528);
// by ID

G=gap.SmallGroup(288,528);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,100,1356,9414]);
// Polycyclic

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

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