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## G = C3×Dic3⋊D4order 288 = 25·32

### Direct product of C3 and Dic3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×Dic3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×Dic3⋊D4
 Lower central C3 — C2×C6 — C3×Dic3⋊D4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×Dic3⋊D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b3c, ce=ec, ede=d-1 >

Subgroups: 546 in 205 conjugacy classes, 66 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×3], C6 [×6], C6 [×10], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], Dic3, C12 [×9], D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×18], C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×3], C3×C6 [×3], C3×C6, C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×6], C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3 [×2], C3×Dic3, C3×C12 [×2], S3×C6 [×2], S3×C6 [×5], C62, C62 [×3], Dic3⋊C4, D6⋊C4, C3×C22⋊C4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], C22×C12, C6×D4 [×3], S3×C12 [×2], C3×D12 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×4], C6×C12 [×2], S3×C2×C6 [×2], C2×C62, Dic3⋊D4, C3×C4⋊D4, C3×Dic3⋊C4, C3×D6⋊C4, C32×C22⋊C4, S3×C2×C12, C6×D12, C6×C3⋊D4 [×2], C3×Dic3⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×4], C23, D6 [×3], C2×C6 [×7], C2×D4 [×2], C4○D4, C3×S3, C3×D4 [×4], C22×S3, C22×C6, C4⋊D4, S3×C6 [×3], C4○D12, S3×D4 [×2], C6×D4 [×2], C3×C4○D4, S3×C2×C6, Dic3⋊D4, C3×C4⋊D4, C3×C4○D12, C3×S3×D4 [×2], C3×Dic3⋊D4

Smallest permutation representation of C3×Dic3⋊D4
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(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 10 4 7)(2 9 5 12)(3 8 6 11)(13 45 16 48)(14 44 17 47)(15 43 18 46)(19 37 22 40)(20 42 23 39)(21 41 24 38)(25 31 28 34)(26 36 29 33)(27 35 30 32)
(1 35 14 40)(2 34 15 39)(3 33 16 38)(4 32 17 37)(5 31 18 42)(6 36 13 41)(7 30 47 19)(8 29 48 24)(9 28 43 23)(10 27 44 22)(11 26 45 21)(12 25 46 20)
(1 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 6P ··· 6W 6X 6Y 6Z 6AA 6AB 6AC 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V 12W 12X order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 size 1 1 1 1 4 6 6 12 1 1 2 2 2 2 2 4 6 6 12 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 12 12 2 2 2 2 4 ··· 4 6 6 6 6 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C4○D4 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S3×D4 C3×S3×D4 kernel C3×Dic3⋊D4 C3×Dic3⋊C4 C3×D6⋊C4 C32×C22⋊C4 S3×C2×C12 C6×D12 C6×C3⋊D4 Dic3⋊D4 Dic3⋊C4 D6⋊C4 C3×C22⋊C4 S3×C2×C4 C2×D12 C2×C3⋊D4 C3×C22⋊C4 C3×Dic3 S3×C6 C2×C12 C22×C6 C3×C6 C22⋊C4 Dic3 D6 C2×C4 C23 C6 C6 C2 C6 C2 # reps 1 1 1 1 1 1 2 2 2 2 2 2 2 4 1 2 2 2 1 2 2 4 4 4 2 4 4 8 2 4

Matrix representation of C3×Dic3⋊D4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 0 5 0 0 5 0
,
 0 1 0 0 12 0 0 0 0 0 0 1 0 0 12 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×Dic3⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic3:D4");
// GroupNames label

G:=SmallGroup(288,655);
// by ID

G=gap.SmallGroup(288,655);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,1598,555,9414]);
// Polycyclic

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

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