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## G = (C4×C12)⋊C6order 288 = 25·32

### 3rd semidirect product of C4×C12 and C6 acting faithfully

Aliases: (C4×C12)⋊3C6, C4⋊D12⋊C3, C42⋊C33S3, C423(C3×S3), C3⋊(C23.A4), C22.1(S3×A4), (C22×S3).1A4, (C3×C42⋊C3)⋊3C2, (C2×C6).1(C2×A4), SmallGroup(288,405)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C12 — (C4×C12)⋊C6
 Chief series C1 — C22 — C2×C6 — C4×C12 — C3×C42⋊C3 — (C4×C12)⋊C6
 Lower central C4×C12 — (C4×C12)⋊C6
 Upper central C1

Generators and relations for (C4×C12)⋊C6
G = < a,b,c | a4=b12=c6=1, ab=ba, cac-1=ab9, cbc-1=a-1b8 >

Subgroups: 498 in 59 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C12, A4, D6, C2×C6, C42, C2×D4, C3×S3, D12, C2×C12, C2×A4, C22×S3, C22×S3, C41D4, C3×A4, C42⋊C3, C42⋊C3, C4×C12, C2×D12, S3×A4, C23.A4, C4⋊D12, C3×C42⋊C3, (C4×C12)⋊C6
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, S3×A4, C23.A4, (C4×C12)⋊C6

Character table of (C4×C12)⋊C6

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 12A 12B 12C 12D size 1 3 12 36 2 16 16 32 32 6 6 6 48 48 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ65 ζ6 1 1 1 1 linear of order 6 ρ4 1 1 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ6 ζ65 1 1 1 1 linear of order 6 ρ5 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 1 1 linear of order 3 ρ6 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 1 1 linear of order 3 ρ7 2 2 0 0 -1 2 2 -1 -1 2 2 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 2 2 -1 0 0 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 2 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 2 2 -1 0 0 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 3 3 -3 1 3 0 0 0 0 -1 -1 3 0 0 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ11 3 3 3 -1 3 0 0 0 0 -1 -1 3 0 0 -1 -1 -1 -1 orthogonal lifted from A4 ρ12 6 -2 0 0 6 0 0 0 0 -2 2 -2 0 0 -2 2 2 -2 orthogonal lifted from C23.A4 ρ13 6 -2 0 0 6 0 0 0 0 2 -2 -2 0 0 2 -2 -2 2 orthogonal lifted from C23.A4 ρ14 6 6 0 0 -3 0 0 0 0 -2 -2 -3 0 0 1 1 1 1 orthogonal lifted from S3×A4 ρ15 6 -2 0 0 -3 0 0 0 0 -2 2 1 0 0 1-2√3 -1 -1 1+2√3 orthogonal faithful ρ16 6 -2 0 0 -3 0 0 0 0 2 -2 1 0 0 -1 1+2√3 1-2√3 -1 orthogonal faithful ρ17 6 -2 0 0 -3 0 0 0 0 -2 2 1 0 0 1+2√3 -1 -1 1-2√3 orthogonal faithful ρ18 6 -2 0 0 -3 0 0 0 0 2 -2 1 0 0 -1 1-2√3 1+2√3 -1 orthogonal faithful

Smallest permutation representation of (C4×C12)⋊C6
On 36 points
Generators in S36
(1 4 11 7)(2 5 12 8)(3 6 10 9)(13 16 19 22)(14 17 20 23)(15 18 21 24)
(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)
(1 22 28 8 15 29)(2 18 32 7 19 25)(3 14 36 9 23 33)(4 13 31 12 24 26)(5 21 35 11 16 34)(6 17 27 10 20 30)

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

G:=Group( (1,4,11,7)(2,5,12,8)(3,6,10,9)(13,16,19,22)(14,17,20,23)(15,18,21,24), (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), (1,22,28,8,15,29)(2,18,32,7,19,25)(3,14,36,9,23,33)(4,13,31,12,24,26)(5,21,35,11,16,34)(6,17,27,10,20,30) );

G=PermutationGroup([[(1,4,11,7),(2,5,12,8),(3,6,10,9),(13,16,19,22),(14,17,20,23),(15,18,21,24)], [(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)], [(1,22,28,8,15,29),(2,18,32,7,19,25),(3,14,36,9,23,33),(4,13,31,12,24,26),(5,21,35,11,16,34),(6,17,27,10,20,30)]])

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

 1 0 0 0 0 0 0 1 0 0 0 0 11 2 9 1 0 0 7 6 9 4 0 0 4 0 10 2 10 7 1 1 0 2 6 3
,
 6 3 0 0 0 0 10 3 0 0 0 0 4 9 1 11 0 0 9 9 8 11 0 0 3 12 7 9 7 10 8 12 2 9 3 10
,
 3 8 3 0 6 12 4 9 0 0 6 7 10 7 0 0 0 0 11 0 6 0 0 10 2 0 7 7 5 10 5 0 11 3 5 9

G:=sub<GL(6,GF(13))| [1,0,11,7,4,1,0,1,2,6,0,1,0,0,9,9,10,0,0,0,1,4,2,2,0,0,0,0,10,6,0,0,0,0,7,3],[6,10,4,9,3,8,3,3,9,9,12,12,0,0,1,8,7,2,0,0,11,11,9,9,0,0,0,0,7,3,0,0,0,0,10,10],[3,4,10,11,2,5,8,9,7,0,0,0,3,0,0,6,7,11,0,0,0,0,7,3,6,6,0,0,5,5,12,7,0,10,10,9] >;

(C4×C12)⋊C6 in GAP, Magma, Sage, TeX

(C_4\times C_{12})\rtimes C_6
% in TeX

G:=Group("(C4xC12):C6");
// GroupNames label

G:=SmallGroup(288,405);
// by ID

G=gap.SmallGroup(288,405);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-2,2,4664,198,772,4371,2110,360,1684,3036,5305]);
// Polycyclic

G:=Group<a,b,c|a^4=b^12=c^6=1,a*b=b*a,c*a*c^-1=a*b^9,c*b*c^-1=a^-1*b^8>;
// generators/relations

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