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G = C4×C6.D6  order 288 = 25·32

Direct product of C4 and C6.D6

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C4×C6.D6, C62.52C23, C12⋊7(C4×S3), Dic32⋊25C2, C3⋊S3⋊1C42, C3⋊1(S3×C42), Dic3⋊7(C4×S3), C32⋊3(C2×C42), (C4×Dic3)⋊17S3, (C2×C12).306D6, (Dic3×C12)⋊25C2, (C6×C12).230C22, (C2×Dic3).112D6, (C6×Dic3).154C22, C2.3(C4×S32), (C4×C3⋊S3)⋊6C4, C6.32(S3×C2×C4), (C2×C4).139S32, (C3×C12)⋊16(C2×C4), C22.32(C2×S32), C3⋊Dic3⋊10(C2×C4), (C3×Dic3)⋊9(C2×C4), C2.2(C2×C6.D6), (C3×C6).54(C22×C4), (C2×C6).71(C22×S3), (C2×C6.D6).9C2, (C22×C3⋊S3).69C22, (C2×C3⋊Dic3).129C22, (C2×C4×C3⋊S3).19C2, (C2×C3⋊S3).29(C2×C4), SmallGroup(288,530)

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 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E···4T4U4V4W4X6A···6F6G6H6I12A···12H12I12J12K12L12M···12AB
order1222222233344444···444446···666612···121212121212···12
size1111999922411113···399992···24442···244446···6

72 irreducible representations

dim1111111222224444
type+++++++++++
imageC1C2C2C2C2C4C4S3D6D6C4×S3C4×S3S32C6.D6C2×S32C4×S32
kernelC4×C6.D6Dic32Dic3×C12C2×C6.D6C2×C4×C3⋊S3C6.D6C4×C3⋊S3C4×Dic3C2×Dic3C2×C12Dic3C12C2×C4C4C22C2
# reps122211682421681214

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

500000
050000
001000
000100
000010
000001
,
1120000
100000
0011200
001000
000010
000001
,
850000
050000
008500
000500
0000012
0000112
,
1210000
010000
0011200
0001200
0000112
0000012

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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