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## G = Dic3⋊D12order 288 = 25·32

### 1st semidirect product of Dic3 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3⋊D12
G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 922 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×7], C6 [×6], C6 [×6], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×2], Dic3 [×4], C12 [×6], D6 [×2], D6 [×15], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×3], C3⋊S3, C3×C6 [×3], C4×S3 [×2], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×5], C2×C3⋊S3 [×3], C62, Dic3⋊C4, D6⋊C4 [×4], C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12 [×2], C22×Dic3, C2×C3⋊D4 [×2], C6×D4, S3×Dic3 [×2], C3⋊D12 [×4], C3×D12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, C12⋊D4, C23.14D6, D6⋊Dic3, C3×Dic3⋊C4, C6.11D12, C2×S3×Dic3, C2×C3⋊D12 [×2], C6×D12, Dic3⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12, S3×D4 [×2], D42S3, Q83S3, C2×C3⋊D4, C2×S32, C12⋊D4, C23.14D6, D12⋊S3, S3×D12, S3×C3⋊D4, Dic3⋊D12

Smallest permutation representation of Dic3⋊D12
On 48 points
Generators in S48
(1 42 5 46 9 38)(2 43 6 47 10 39)(3 44 7 48 11 40)(4 45 8 37 12 41)(13 25 21 33 17 29)(14 26 22 34 18 30)(15 27 23 35 19 31)(16 28 24 36 20 32)
(1 20 46 28)(2 29 47 21)(3 22 48 30)(4 31 37 23)(5 24 38 32)(6 33 39 13)(7 14 40 34)(8 35 41 15)(9 16 42 36)(10 25 43 17)(11 18 44 26)(12 27 45 19)
(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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 12 36 2 2 4 4 6 6 12 18 18 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C4○D4 D12 C3⋊D4 S32 S3×D4 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 S3×D12 S3×C3⋊D4 kernel Dic3⋊D12 D6⋊Dic3 C3×Dic3⋊C4 C6.11D12 C2×S3×Dic3 C2×C3⋊D12 C6×D12 Dic3⋊C4 C2×D12 C3×Dic3 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 D6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 1 2 2 2 2 2 2 4 4 1 2 1 1 1 2 2 2

Matrix representation of Dic3⋊D12 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 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 -1 0 0 0 0 0 0 0 0 -1
,
 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 0 -1 -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 0 0 0 -1 0 0 0 0 0 0 1 0
,
 0 -1 0 0 0 0 0 0 1 0 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 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

Dic3⋊D12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes D_{12}
% in TeX

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

G:=SmallGroup(288,534);
// by ID

G=gap.SmallGroup(288,534);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,58,1356,9414]);
// Polycyclic

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

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