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## G = C4×C6.D6order 288 = 25·32

### Direct product of C4 and C6.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C6.D6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C4×C6.D6
 Lower central C32 — C4×C6.D6
 Upper central C1 — C2×C4

Generators and relations for C4×C6.D6
G = < a,b,c,d | a4=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 786 in 243 conjugacy classes, 92 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×10], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×17], C23, C32, Dic3 [×8], Dic3 [×6], C12 [×4], C12 [×10], D6 [×18], C2×C6 [×2], C2×C6, C42 [×4], C22×C4 [×3], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×28], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C2×C42, C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C4×Dic3 [×2], C4×Dic3 [×4], C4×C12 [×2], S3×C2×C4 [×7], C6.D6 [×8], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C42 [×2], Dic32 [×2], Dic3×C12 [×2], C2×C6.D6 [×2], C2×C4×C3⋊S3, C4×C6.D6
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3 [×2], C2×C4 [×18], C23, D6 [×6], C42 [×4], C22×C4 [×3], C4×S3 [×12], C22×S3 [×2], C2×C42, S32, S3×C2×C4 [×6], C6.D6 [×2], C2×S32, S3×C42 [×2], C4×S32 [×2], C2×C6.D6, C4×C6.D6

Smallest permutation representation of C4×C6.D6
On 48 points
Generators in S48
(1 19 38 30)(2 20 39 31)(3 21 40 32)(4 22 41 33)(5 23 42 34)(6 24 43 35)(7 13 44 36)(8 14 45 25)(9 15 46 26)(10 16 47 27)(11 17 48 28)(12 18 37 29)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 24)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 39 41 43 45 47)(38 48 46 44 42 40)
(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 46)(2 39)(3 44)(4 37)(5 42)(6 47)(7 40)(8 45)(9 38)(10 43)(11 48)(12 41)(13 32)(14 25)(15 30)(16 35)(17 28)(18 33)(19 26)(20 31)(21 36)(22 29)(23 34)(24 27)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E ··· 4T 4U 4V 4W 4X 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I 12J 12K 12L 12M ··· 12AB order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 4 1 1 1 1 3 ··· 3 9 9 9 9 2 ··· 2 4 4 4 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 S3 D6 D6 C4×S3 C4×S3 S32 C6.D6 C2×S32 C4×S32 kernel C4×C6.D6 Dic32 Dic3×C12 C2×C6.D6 C2×C4×C3⋊S3 C6.D6 C4×C3⋊S3 C4×Dic3 C2×Dic3 C2×C12 Dic3 C12 C2×C4 C4 C22 C2 # reps 1 2 2 2 1 16 8 2 4 2 16 8 1 2 1 4

Matrix representation of C4×C6.D6 in GL6(𝔽13)

 5 0 0 0 0 0 0 5 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 1
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 5 0 0 0 0 0 5 0 0 0 0 0 0 8 5 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 1 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 12

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

C4×C6.D6 in GAP, Magma, Sage, TeX

C_4\times C_6.D_6
% in TeX

G:=Group("C4xC6.D6");
// GroupNames label

G:=SmallGroup(288,530);
// by ID

G=gap.SmallGroup(288,530);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^6=d^2=1,c^6=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^5>;
// generators/relations

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