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## G = C3×C42⋊C6order 288 = 25·32

### Direct product of C3 and C42⋊C6

Aliases: C3×C42⋊C6, (C4×C12)⋊2C6, C42⋊C31C6, C421(C3×C6), C422C2⋊C32, C23.1(C3×A4), C22.3(C6×A4), (C22×C6).5A4, (C3×C42⋊C3)⋊2C2, (C3×C422C2)⋊C3, (C2×C6).11(C2×A4), SmallGroup(288,635)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C42⋊C6
G = < a,b,c,d | a3=b4=c4=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=c-1, dcd-1=b-1c >

Character table of C3×C42⋊C6

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D 12E 12F size 1 3 4 1 1 16 16 16 16 16 16 6 6 12 3 3 4 4 16 16 16 16 16 16 6 6 6 6 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 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 linear of order 3 ρ4 1 1 -1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 -1 ζ3 ζ32 ζ6 ζ65 -1 ζ6 -1 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 linear of order 6 ρ5 1 1 1 ζ3 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 linear of order 3 ρ7 1 1 1 ζ32 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ8 1 1 -1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 -1 ζ32 ζ3 ζ65 ζ6 -1 ζ65 -1 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 linear of order 6 ρ9 1 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ10 1 1 -1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 -1 1 1 -1 -1 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 1 1 1 1 -1 -1 linear of order 6 ρ11 1 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ12 1 1 -1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 linear of order 6 ρ13 1 1 -1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 linear of order 6 ρ14 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ15 1 1 -1 ζ3 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 1 1 -1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ6 ζ65 -1 ζ65 -1 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 linear of order 6 ρ16 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ17 1 1 -1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 -1 1 1 -1 -1 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 1 1 1 1 -1 -1 linear of order 6 ρ18 1 1 -1 ζ32 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 1 1 -1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ65 ζ6 -1 ζ6 -1 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 linear of order 6 ρ19 3 3 -3 3 3 0 0 0 0 0 0 -1 -1 1 3 3 -3 -3 0 0 0 0 0 0 -1 -1 -1 -1 1 1 orthogonal lifted from C2×A4 ρ20 3 3 3 3 3 0 0 0 0 0 0 -1 -1 -1 3 3 3 3 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ21 3 3 -3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 1 -3-3√-3/2 -3+3√-3/2 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ65 ζ3 ζ32 complex lifted from C6×A4 ρ22 3 3 -3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 1 -3+3√-3/2 -3-3√-3/2 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ6 ζ32 ζ3 complex lifted from C6×A4 ρ23 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 complex lifted from C3×A4 ρ24 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 complex lifted from C3×A4 ρ25 6 -2 0 6 6 0 0 0 0 0 0 -2i 2i 0 -2 -2 0 0 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 complex lifted from C42⋊C6 ρ26 6 -2 0 6 6 0 0 0 0 0 0 2i -2i 0 -2 -2 0 0 0 0 0 0 0 0 2i -2i 2i -2i 0 0 complex lifted from C42⋊C6 ρ27 6 -2 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 -2i 2i 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 2ζ43ζ3 2ζ4ζ3 2ζ43ζ32 2ζ4ζ32 0 0 complex faithful ρ28 6 -2 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 2i -2i 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 2ζ4ζ32 2ζ43ζ32 2ζ4ζ3 2ζ43ζ3 0 0 complex faithful ρ29 6 -2 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 2i -2i 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 2ζ4ζ3 2ζ43ζ3 2ζ4ζ32 2ζ43ζ32 0 0 complex faithful ρ30 6 -2 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 -2i 2i 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 2ζ43ζ32 2ζ4ζ32 2ζ43ζ3 2ζ4ζ3 0 0 complex faithful

Smallest permutation representation of C3×C42⋊C6
On 48 points
Generators in S48
(1 2 3)(4 8 11)(5 9 12)(6 7 10)(13 19 29)(14 20 30)(15 21 25)(16 22 26)(17 23 27)(18 24 28)(31 37 47)(32 38 48)(33 39 43)(34 40 44)(35 41 45)(36 42 46)
(1 25 7 38)(2 15 10 48)(3 21 6 32)(4 41 12 28)(5 18 8 45)(9 24 11 35)(13 40 16 30)(14 19 44 22)(17 36 47 33)(20 29 34 26)(23 42 31 39)(27 46 37 43)
(1 47 12 14)(2 31 5 20)(3 37 9 30)(4 44 7 17)(6 27 11 40)(8 34 10 23)(13 21 43 24)(15 39 18 29)(16 32 46 35)(19 25 33 28)(22 38 36 41)(26 48 42 45)
(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)

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

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

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

Matrix representation of C3×C42⋊C6 in GL9(𝔽13)

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

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

C3×C42⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_6
% in TeX

G:=Group("C3xC4^2:C6");
// GroupNames label

G:=SmallGroup(288,635);
// by ID

G=gap.SmallGroup(288,635);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,2523,514,360,6304,3476,102,3036,5305]);
// Polycyclic

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

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