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## G = C3×D6⋊3D4order 288 = 25·32

### Direct product of C3 and D6⋊3D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×D6⋊3D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×D6⋊3D4
 Lower central C3 — C2×C6 — C3×D6⋊3D4
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×D63D4
G = < a,b,c,d,e | a3=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 522 in 215 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×2], C6 [×6], C6 [×13], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C32, Dic3 [×3], C12 [×4], C12 [×5], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×25], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×6], C3×D4 [×12], C22×S3, C22×C6 [×4], C22×C6 [×3], C4⋊D4, C3×Dic3 [×3], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×6], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×C3⋊D4 [×2], C22×C12, C6×D4 [×2], C6×D4 [×3], S3×C12 [×2], C6×Dic3, C6×Dic3 [×2], C3×C3⋊D4 [×4], C6×C12, D4×C32 [×2], S3×C2×C6, C2×C62 [×2], D63D4, C3×C4⋊D4, C3×C4⋊Dic3, C3×C6.D4 [×2], S3×C2×C12, C6×C3⋊D4 [×2], D4×C3×C6, C3×D63D4
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 [×2], C3×D4 [×4], C22×S3, C22×C6, C4⋊D4, S3×C6 [×3], S3×D4, D42S3, C2×C3⋊D4, C6×D4 [×2], C3×C4○D4, C3×C3⋊D4 [×2], S3×C2×C6, D63D4, C3×C4⋊D4, C3×S3×D4, C3×D42S3, C6×C3⋊D4, C3×D63D4

Smallest permutation representation of C3×D63D4
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 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 47)(26 46)(27 45)(28 44)(29 43)(30 48)
(1 20 16 27)(2 21 17 28)(3 22 18 29)(4 23 13 30)(5 24 14 25)(6 19 15 26)(7 31 43 38)(8 32 44 39)(9 33 45 40)(10 34 46 41)(11 35 47 42)(12 36 48 37)
(7 46)(8 47)(9 48)(10 43)(11 44)(12 45)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)

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

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

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

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 ··· 6AE 6AF 6AG 6AH 6AI 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R 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 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 4 4 6 6 1 1 2 2 2 2 2 6 6 12 12 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

72 irreducible representations

 dim 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 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C4○D4 C3×S3 C3⋊D4 C3×D4 C3×D4 S3×C6 S3×C6 C3×C4○D4 C3×C3⋊D4 S3×D4 D4⋊2S3 C3×S3×D4 C3×D4⋊2S3 kernel C3×D6⋊3D4 C3×C4⋊Dic3 C3×C6.D4 S3×C2×C12 C6×C3⋊D4 D4×C3×C6 D6⋊3D4 C4⋊Dic3 C6.D4 S3×C2×C4 C2×C3⋊D4 C6×D4 C6×D4 C3×C12 S3×C6 C2×C12 C22×C6 C3×C6 C2×D4 C12 C12 D6 C2×C4 C23 C6 C4 C6 C6 C2 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 2 2 1 2 2 2 4 4 4 2 4 4 8 1 1 2 2

Matrix representation of C3×D63D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 10 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 5 0 0 0 0 0 0 0 4 5 0 0 0 0 10 9
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,10,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,4,10,0,0,0,0,5,9],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×D63D4 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes_3D_4
% in TeX

G:=Group("C3xD6:3D4");
// GroupNames label

G:=SmallGroup(288,709);
// by ID

G=gap.SmallGroup(288,709);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,9414]);
// Polycyclic

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

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