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## G = Dic3×C3⋊D4order 288 = 25·32

### Direct product of Dic3 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — Dic3×C3⋊D4
 Lower central C32 — C3×C6 — Dic3×C3⋊D4
 Upper central C1 — C22 — C23

Generators and relations for Dic3×C3⋊D4
G = < a,b,c,d,e | a6=c3=d4=e2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 634 in 205 conjugacy classes, 72 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×6], C6 [×11], C2×C4 [×9], D4 [×4], C23, C23, C32, Dic3 [×4], Dic3 [×7], C12 [×5], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×13], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×10], C3⋊D4 [×4], C2×C12 [×5], C3×D4 [×4], C22×S3, C22×C6 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×4], C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×2], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4 [×4], S3×C2×C4, C22×Dic3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×Dic3 [×2], C6×Dic3 [×3], C6×Dic3 [×2], C3×C3⋊D4 [×4], C2×C3⋊Dic3 [×2], S3×C2×C6, C2×C62, C4×C3⋊D4, D4×Dic3, Dic32, D6⋊Dic3, Dic3⋊Dic3, C625C4, C2×S3×Dic3, Dic3×C2×C6, C6×C3⋊D4, Dic3×C3⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C2×Dic3 [×6], C3⋊D4 [×2], C22×S3 [×2], C4×D4, S32, S3×C2×C4, C4○D12, S3×D4, D42S3, C22×Dic3, C2×C3⋊D4, S3×Dic3 [×2], C2×S32, C4×C3⋊D4, D4×Dic3, C2×S3×Dic3, D6.3D6, S3×C3⋊D4, Dic3×C3⋊D4

Smallest permutation representation of Dic3×C3⋊D4
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 21 4 24)(2 20 5 23)(3 19 6 22)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 29 16 26)(14 28 17 25)(15 27 18 30)(31 44 34 47)(32 43 35 46)(33 48 36 45)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 39 18 33)(2 40 13 34)(3 41 14 35)(4 42 15 36)(5 37 16 31)(6 38 17 32)(7 30 48 21)(8 25 43 22)(9 26 44 23)(10 27 45 24)(11 28 46 19)(12 29 47 20)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 30)(8 25)(9 26)(10 27)(11 28)(12 29)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

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

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

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 ··· 6J 6K ··· 6S 6T 6U 12A ··· 12H 12I 12J 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 12 ··· 12 12 12 size 1 1 1 1 2 2 6 6 2 2 4 3 3 3 3 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 12 12 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 Dic3 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S32 S3×D4 D4⋊2S3 S3×Dic3 C2×S32 D6.3D6 S3×C3⋊D4 kernel Dic3×C3⋊D4 Dic32 D6⋊Dic3 Dic3⋊Dic3 C62⋊5C4 C2×S3×Dic3 Dic3×C2×C6 C6×C3⋊D4 C3×C3⋊D4 C22×Dic3 C2×C3⋊D4 C3×Dic3 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C3×C6 Dic3 C2×C6 C6 C23 C6 C6 C22 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 3 4 1 2 2 4 4 4 1 1 1 2 1 2 2

Matrix representation of Dic3×C3⋊D4 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 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 12 12 0 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1
,
 12 1 0 0 0 0 0 0 12 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,11,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;

Dic3×C3⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\rtimes D_4
% in TeX

G:=Group("Dic3xC3:D4");
// GroupNames label

G:=SmallGroup(288,620);
// by ID

G=gap.SmallGroup(288,620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^3=d^4=e^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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