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## G = C3×C24⋊C6order 288 = 25·32

### Direct product of C3 and C24⋊C6

Aliases: C3×C24⋊C6, C22⋊A43C6, C231(C3×A4), C22≀C2⋊C32, C242(C3×C6), (C22×C6)⋊1A4, (C23×C6)⋊1C6, C22.2(C6×A4), (C3×C22≀C2)⋊C3, (C3×C22⋊A4)⋊1C2, (C2×C6).10(C2×A4), SmallGroup(288,634)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C24⋊C6
 Chief series C1 — C22 — C24 — C23×C6 — C3×C22⋊A4 — C3×C24⋊C6
 Lower central C24 — C3×C24⋊C6
 Upper central C1 — C3

Generators and relations for C3×C24⋊C6
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f6=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=ec=ce, cd=dc, fcf-1=bcde, fef-1=de=ed, fdf-1=e >

Subgroups: 444 in 86 conjugacy classes, 18 normal (12 characteristic)
C1, C2 [×4], C3, C3 [×3], C4, C22, C22 [×9], C6 [×7], C2×C4, D4 [×2], C23, C23 [×3], C32, C12, A4 [×9], C2×C6, C2×C6 [×9], C22⋊C4, C2×D4, C24, C3×C6, C2×C12, C3×D4 [×2], C2×A4 [×3], C22×C6, C22×C6 [×3], C22≀C2, C3×A4 [×3], C3×C22⋊C4, C6×D4, C22⋊A4 [×3], C23×C6, C6×A4, C24⋊C6 [×3], C3×C22≀C2, C3×C22⋊A4, C3×C24⋊C6
Quotients: C1, C2, C3 [×4], C6 [×4], C32, A4, C3×C6, C2×A4, C3×A4, C6×A4, C24⋊C6, C3×C24⋊C6

Character table of C3×C24⋊C6

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 3F 3G 3H 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 12A 12B size 1 3 4 6 6 1 1 16 16 16 16 16 16 12 3 3 4 4 6 6 6 6 16 16 16 16 16 16 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 -1 ζ3 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 ζ65 -1 ζ6 -1 ζ6 ζ65 linear of order 6 ρ4 1 1 -1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 -1 1 1 -1 -1 1 1 1 1 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 -1 -1 linear of order 6 ρ5 1 1 1 1 1 ζ3 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 linear of order 3 ρ8 1 1 1 1 1 ζ32 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 linear of order 3 ρ9 1 1 -1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 -1 ζ32 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 -1 -1 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 linear of order 6 ρ10 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ11 1 1 1 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 linear of order 3 ρ12 1 1 -1 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 -1 ζ32 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 ζ6 -1 ζ65 -1 ζ65 ζ6 linear of order 6 ρ13 1 1 -1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 -1 1 1 -1 -1 1 1 1 1 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 -1 -1 linear of order 6 ρ14 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 linear of order 3 ρ15 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ16 1 1 -1 1 1 ζ3 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 -1 ζ3 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 -1 ζ65 -1 ζ6 ζ6 ζ65 linear of order 6 ρ17 1 1 -1 1 1 ζ32 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 -1 ζ32 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 -1 ζ6 -1 ζ65 ζ65 ζ6 linear of order 6 ρ18 1 1 -1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 -1 ζ3 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 -1 -1 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 linear of order 6 ρ19 3 3 3 -1 -1 3 3 0 0 0 0 0 0 -1 3 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 orthogonal lifted from A4 ρ20 3 3 -3 -1 -1 3 3 0 0 0 0 0 0 1 3 3 -3 -3 -1 -1 -1 -1 0 0 0 0 0 0 1 1 orthogonal lifted from C2×A4 ρ21 3 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 ζ65 ζ65 ζ6 ζ6 0 0 0 0 0 0 ζ6 ζ65 complex lifted from C3×A4 ρ22 3 3 -3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 1 -3-3√-3/2 -3+3√-3/2 3+3√-3/2 3-3√-3/2 ζ6 ζ6 ζ65 ζ65 0 0 0 0 0 0 ζ3 ζ32 complex lifted from C6×A4 ρ23 3 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 ζ6 ζ6 ζ65 ζ65 0 0 0 0 0 0 ζ65 ζ6 complex lifted from C3×A4 ρ24 3 3 -3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 1 -3+3√-3/2 -3-3√-3/2 3-3√-3/2 3+3√-3/2 ζ65 ζ65 ζ6 ζ6 0 0 0 0 0 0 ζ32 ζ3 complex lifted from C6×A4 ρ25 6 -2 0 -2 2 6 6 0 0 0 0 0 0 0 -2 -2 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C6 ρ26 6 -2 0 2 -2 6 6 0 0 0 0 0 0 0 -2 -2 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C6 ρ27 6 -2 0 2 -2 -3-3√-3 -3+3√-3 0 0 0 0 0 0 0 1+√-3 1-√-3 0 0 -1-√-3 1+√-3 -1+√-3 1-√-3 0 0 0 0 0 0 0 0 complex faithful ρ28 6 -2 0 2 -2 -3+3√-3 -3-3√-3 0 0 0 0 0 0 0 1-√-3 1+√-3 0 0 -1+√-3 1-√-3 -1-√-3 1+√-3 0 0 0 0 0 0 0 0 complex faithful ρ29 6 -2 0 -2 2 -3-3√-3 -3+3√-3 0 0 0 0 0 0 0 1+√-3 1-√-3 0 0 1+√-3 -1-√-3 1-√-3 -1+√-3 0 0 0 0 0 0 0 0 complex faithful ρ30 6 -2 0 -2 2 -3+3√-3 -3-3√-3 0 0 0 0 0 0 0 1-√-3 1+√-3 0 0 1-√-3 -1+√-3 1+√-3 -1-√-3 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×C24⋊C6
On 24 points - transitive group 24T698
Generators in S24
(1 6 4)(2 5 3)(7 14 21)(8 15 22)(9 16 23)(10 17 24)(11 18 19)(12 13 20)
(1 10)(4 24)(6 17)(8 12)(13 15)(20 22)
(1 12)(4 20)(6 13)(8 10)(15 17)(22 24)
(1 10)(2 7)(3 21)(4 24)(5 14)(6 17)(8 12)(9 11)(13 15)(16 18)(19 23)(20 22)
(1 12)(2 9)(3 23)(4 20)(5 16)(6 13)(7 11)(8 10)(14 18)(15 17)(19 21)(22 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)

G:=sub<Sym(24)| (1,6,4)(2,5,3)(7,14,21)(8,15,22)(9,16,23)(10,17,24)(11,18,19)(12,13,20), (1,10)(4,24)(6,17)(8,12)(13,15)(20,22), (1,12)(4,20)(6,13)(8,10)(15,17)(22,24), (1,10)(2,7)(3,21)(4,24)(5,14)(6,17)(8,12)(9,11)(13,15)(16,18)(19,23)(20,22), (1,12)(2,9)(3,23)(4,20)(5,16)(6,13)(7,11)(8,10)(14,18)(15,17)(19,21)(22,24), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;

G:=Group( (1,6,4)(2,5,3)(7,14,21)(8,15,22)(9,16,23)(10,17,24)(11,18,19)(12,13,20), (1,10)(4,24)(6,17)(8,12)(13,15)(20,22), (1,12)(4,20)(6,13)(8,10)(15,17)(22,24), (1,10)(2,7)(3,21)(4,24)(5,14)(6,17)(8,12)(9,11)(13,15)(16,18)(19,23)(20,22), (1,12)(2,9)(3,23)(4,20)(5,16)(6,13)(7,11)(8,10)(14,18)(15,17)(19,21)(22,24), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );

G=PermutationGroup([(1,6,4),(2,5,3),(7,14,21),(8,15,22),(9,16,23),(10,17,24),(11,18,19),(12,13,20)], [(1,10),(4,24),(6,17),(8,12),(13,15),(20,22)], [(1,12),(4,20),(6,13),(8,10),(15,17),(22,24)], [(1,10),(2,7),(3,21),(4,24),(5,14),(6,17),(8,12),(9,11),(13,15),(16,18),(19,23),(20,22)], [(1,12),(2,9),(3,23),(4,20),(5,16),(6,13),(7,11),(8,10),(14,18),(15,17),(19,21),(22,24)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)])

G:=TransitiveGroup(24,698);

Matrix representation of C3×C24⋊C6 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 0 12 1 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 12 0 0 0 1 0 12 0 0 0 0 0 12
,
 0 1 12 0 0 0 1 0 12 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 0 1 0 0 0 12 1 0
,
 12 0 0 0 0 0 12 0 1 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 1 0 0 0 12 1 0
,
 0 12 1 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 1 12 0
,
 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0

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

C3×C24⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes C_6
% in TeX

G:=Group("C3xC2^4:C6");
// GroupNames label

G:=SmallGroup(288,634);
// by ID

G=gap.SmallGroup(288,634);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,2523,514,6304,956,3036,5305]);
// Polycyclic

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

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