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## G = Dic12⋊S3order 288 = 25·32

### 2nd semidirect product of Dic12 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — Dic12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×Dic6 — Dic12⋊S3
 Lower central C32 — C3×C6 — C3×C12 — Dic12⋊S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for Dic12⋊S3
G = < a,b,c,d | a24=c3=d2=1, b2=a12, bab-1=a-1, ac=ca, dad=a13, bc=cb, bd=db, dcd=c-1 >

Subgroups: 586 in 130 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×4], C6 [×2], C6 [×2], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3, Dic3 [×5], C12 [×2], C12 [×4], D6, D6 [×3], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], Dic6 [×5], C4×S3, C4×S3 [×3], D12 [×4], C2×Dic3, C3⋊D4, C2×C12, C3×Q8 [×2], C8.C22, C3×Dic3, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2 [×4], Dic12, Dic12, Q82S3, C3⋊Q16, C3×M4(2), C3×Q16, C2×Dic6, C4○D12, S3×Q8, Q83S3, C3×C3⋊C8, C3×C24, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C3×Dic6 [×2], S3×C12, C324Q8, C12⋊S3, C8.D6, Q16⋊S3, C325SD16, C323Q16, C3×C8⋊S3, C3×Dic12, C242S3, S3×Dic6, D6.6D6, Dic12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C8.C22, S32, C2×D12, S3×D4, C2×S32, C8.D6, Q16⋊S3, S3×D12, Dic12⋊S3

Smallest permutation representation of Dic12⋊S3
On 48 points
Generators in S48
(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 45 13 33)(2 44 14 32)(3 43 15 31)(4 42 16 30)(5 41 17 29)(6 40 18 28)(7 39 19 27)(8 38 20 26)(9 37 21 25)(10 36 22 48)(11 35 23 47)(12 34 24 46)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 42)(2 31)(3 44)(4 33)(5 46)(6 35)(7 48)(8 37)(9 26)(10 39)(11 28)(12 41)(13 30)(14 43)(15 32)(16 45)(17 34)(18 47)(19 36)(20 25)(21 38)(22 27)(23 40)(24 29)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H 24I 24J order 1 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 8 8 12 12 12 12 12 12 12 12 24 ··· 24 24 24 size 1 1 6 36 2 2 4 2 6 12 12 36 2 2 4 12 4 12 2 2 4 4 4 12 24 24 4 ··· 4 12 12

36 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 4 4 4 type + + + + + + + + + + + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D12 D12 C8.C22 S32 S3×D4 C2×S32 C8.D6 Q16⋊S3 S3×D12 Dic12⋊S3 kernel Dic12⋊S3 C32⋊5SD16 C32⋊3Q16 C3×C8⋊S3 C3×Dic12 C24⋊2S3 S3×Dic6 D6.6D6 C8⋊S3 Dic12 C3×Dic3 S3×C6 C3⋊C8 C24 Dic6 C4×S3 Dic3 D6 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 2 2 1 1 1 1 2 2 2 4

Matrix representation of Dic12⋊S3 in GL4(𝔽73) generated by

 36 25 0 0 48 11 0 0 0 0 62 25 0 0 48 37
,
 0 0 1 0 0 0 0 1 72 0 0 0 0 72 0 0
,
 72 72 0 0 1 0 0 0 0 0 72 72 0 0 1 0
,
 0 0 48 11 0 0 36 25 25 62 0 0 37 48 0 0
G:=sub<GL(4,GF(73))| [36,48,0,0,25,11,0,0,0,0,62,48,0,0,25,37],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[72,1,0,0,72,0,0,0,0,0,72,1,0,0,72,0],[0,0,25,37,0,0,62,48,48,36,0,0,11,25,0,0] >;

Dic12⋊S3 in GAP, Magma, Sage, TeX

{\rm Dic}_{12}\rtimes S_3
% in TeX

G:=Group("Dic12:S3");
// GroupNames label

G:=SmallGroup(288,449);
// by ID

G=gap.SmallGroup(288,449);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,142,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^24=c^3=d^2=1,b^2=a^12,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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