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G = (C6×C12)⋊2C4order 288 = 25·32

2nd semidirect product of C6×C12 and C4 acting faithfully

metabelian, soluble, monomial

Aliases: (C6×C12)⋊2C4, (C3×C6).5C42, C62.9(C2×C4), C2.2(C62⋊C4), C323(C2.C42), (C2×C32⋊C4)⋊3C4, C2.5(C4×C32⋊C4), C3⋊S3.4(C4⋊C4), (C2×C3⋊S3).7Q8, (C2×C4)⋊2(C32⋊C4), (C2×C3⋊Dic3)⋊6C4, (C2×C3⋊S3).50D4, (C3×C6).17(C4⋊C4), C3⋊S3.6(C22⋊C4), C2.3(C4⋊(C32⋊C4)), (C22×C32⋊C4).3C2, C22.13(C2×C32⋊C4), (C3×C6).15(C22⋊C4), (C22×C3⋊S3).62C22, (C2×C4×C3⋊S3).14C2, (C2×C3⋊S3).24(C2×C4), SmallGroup(288,429)

Series: Derived Chief Lower central Upper central

C1C3×C6 — (C6×C12)⋊2C4
C1C32C3⋊S3C2×C3⋊S3C22×C3⋊S3C22×C32⋊C4 — (C6×C12)⋊2C4
C32C3×C6 — (C6×C12)⋊2C4
C1C22C2×C4

Generators and relations for (C6×C12)⋊2C4
 G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b8, cbc-1=a-1b >

Subgroups: 696 in 130 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C4 [×6], C22, C22 [×6], S3 [×8], C6 [×6], C2×C4, C2×C4 [×11], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2.C42, C3⋊Dic3, C3×C12, C32⋊C4 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62, S3×C2×C4 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C2×C32⋊C4 [×4], C2×C32⋊C4 [×4], C22×C3⋊S3, C2×C4×C3⋊S3, C22×C32⋊C4 [×2], (C6×C12)⋊2C4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C32⋊C4, C2×C32⋊C4, C4×C32⋊C4, C4⋊(C32⋊C4), C62⋊C4, (C6×C12)⋊2C4

Smallest permutation representation of (C6×C12)⋊2C4
On 48 points
Generators in S48
(1 12 18 5 15 23)(2 9 19 6 16 24)(3 10 20 7 13 21)(4 11 17 8 14 22)(25 48 33 44 29 40)(26 37 34 45 30 41)(27 38 35 46 31 42)(28 39 36 47 32 43)
(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 32 23 43)(2 48 24 33)(3 26 21 37)(4 42 22 27)(5 39 18 36)(6 29 19 40)(7 45 20 30)(8 35 17 46)(9 25 16 44)(10 41 13 34)(11 31 14 38)(12 47 15 28)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C···4L6A···6F12A···12H
order1222222233444···46···612···12
size11119999442218···184···44···4

36 irreducible representations

dim1111112244444
type++++-+++
imageC1C2C2C4C4C4D4Q8C32⋊C4C2×C32⋊C4C4×C32⋊C4C4⋊(C32⋊C4)C62⋊C4
kernel(C6×C12)⋊2C4C2×C4×C3⋊S3C22×C32⋊C4C2×C3⋊Dic3C6×C12C2×C32⋊C4C2×C3⋊S3C2×C3⋊S3C2×C4C22C2C2C2
# reps1122283122444

Matrix representation of (C6×C12)⋊2C4 in GL6(𝔽13)

1200000
0120000
001100
0012000
000011
0000120
,
5100000
880000
008000
000800
000088
000050
,
490000
790000
000010
000001
0012000
001100

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[5,8,0,0,0,0,10,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[4,7,0,0,0,0,9,9,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C6×C12)⋊2C4 in GAP, Magma, Sage, TeX

(C_6\times C_{12})\rtimes_2C_4
% in TeX

G:=Group("(C6xC12):2C4");
// GroupNames label

G:=SmallGroup(288,429);
// by ID

G=gap.SmallGroup(288,429);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^6=b^12=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^8,c*b*c^-1=a^-1*b>;
// generators/relations

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