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## G = C3×D4×Dic3order 288 = 25·32

### Direct product of C3, D4 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D4×Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C2×C6 — C3×D4×Dic3
 Lower central C3 — C6 — C3×D4×Dic3
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×D4×Dic3
G = < a,b,c,d,e | a3=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 426 in 215 conjugacy classes, 102 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3×C12, C62, C62, C62, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×D4, C6×D4, C6×Dic3, C6×Dic3, C6×Dic3, C6×C12, D4×C32, C2×C62, D4×Dic3, D4×C12, Dic3×C12, C3×C4⋊Dic3, C3×C6.D4, Dic3×C2×C6, D4×C3×C6, C3×D4×Dic3
Quotients:

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A ··· 6F 6G ··· 6W 6X ··· 6AI 12A 12B 12C 12D 12E ··· 12L 12M ··· 12R 12S ··· 12AD order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 2 2 2 2 2 3 3 3 3 6 ··· 6 1 ··· 1 2 ··· 2 4 ··· 4 2 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6

90 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 4 4 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 D4 D6 Dic3 D6 C4○D4 C3×S3 C3×D4 S3×C6 C3×Dic3 S3×C6 C3×C4○D4 S3×D4 D4⋊2S3 C3×S3×D4 C3×D4⋊2S3 kernel C3×D4×Dic3 Dic3×C12 C3×C4⋊Dic3 C3×C6.D4 Dic3×C2×C6 D4×C3×C6 D4×Dic3 D4×C32 C4×Dic3 C4⋊Dic3 C6.D4 C22×Dic3 C6×D4 C3×D4 C6×D4 C3×Dic3 C2×C12 C3×D4 C22×C6 C3×C6 C2×D4 Dic3 C2×C4 D4 C23 C6 C6 C6 C2 C2 # reps 1 1 1 2 2 1 2 8 2 2 4 4 2 16 1 2 1 4 2 2 2 4 2 8 4 4 1 1 2 2

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

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 0 1 0 0 12 0 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 10 0 0 0 0 4
,
 5 0 0 0 0 5 0 0 0 0 0 1 0 0 12 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,10,0,0,0,0,4],[5,0,0,0,0,5,0,0,0,0,0,12,0,0,1,0] >;

C3×D4×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xD4xDic3");
// GroupNames label

G:=SmallGroup(288,705);
// by ID

G=gap.SmallGroup(288,705);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,555,9414]);
// Polycyclic

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

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