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G = C6×C42⋊C3order 288 = 25·32

Direct product of C6 and C42⋊C3

Aliases: C6×C42⋊C3, (C4×C12)⋊6C6, (C2×C42)⋊C32, C423(C3×C6), C23.5(C3×A4), C22.1(C6×A4), (C22×C6).11A4, (C2×C4×C12)⋊C3, (C2×C6).9(C2×A4), SmallGroup(288,632)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C42⋊C3
G = < a,b,c,d | a6=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

Subgroups: 276 in 68 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C12, A4, C2×C6, C2×C6, C42, C42, C22×C4, C3×C6, C2×C12, C2×A4, C22×C6, C2×C42, C3×A4, C42⋊C3, C4×C12, C4×C12, C22×C12, C6×A4, C2×C42⋊C3, C2×C4×C12, C3×C42⋊C3, C6×C42⋊C3
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, C3×A4, C42⋊C3, C6×A4, C2×C42⋊C3, C3×C42⋊C3, C6×C42⋊C3

Smallest permutation representation of C6×C42⋊C3
On 36 points
Generators in S36
(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 4)(2 5)(3 6)(7 35 10 32)(8 36 11 33)(9 31 12 34)(13 16)(14 17)(15 18)(19 30 22 27)(20 25 23 28)(21 26 24 29)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(19 27 22 30)(20 28 23 25)(21 29 24 26)
(1 31 21)(2 32 22)(3 33 23)(4 34 24)(5 35 19)(6 36 20)(7 27 17)(8 28 18)(9 29 13)(10 30 14)(11 25 15)(12 26 16)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A ··· 4H 6A 6B 6C 6D 6E 6F 6G ··· 6L 12A ··· 12P order 1 2 2 2 3 3 3 ··· 3 4 ··· 4 6 6 6 6 6 6 6 ··· 6 12 ··· 12 size 1 1 3 3 1 1 16 ··· 16 3 ··· 3 1 1 3 3 3 3 16 ··· 16 3 ··· 3

48 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 C3×A4 C42⋊C3 C6×A4 C2×C42⋊C3 C3×C42⋊C3 C6×C42⋊C3 kernel C6×C42⋊C3 C3×C42⋊C3 C2×C42⋊C3 C2×C4×C12 C42⋊C3 C4×C12 C22×C6 C2×C6 C23 C6 C22 C3 C2 C1 # reps 1 1 6 2 6 2 1 1 2 4 2 4 8 8

Matrix representation of C6×C42⋊C3 in GL3(𝔽13) generated by

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

C6×C42⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("C6xC4^2:C3");
// GroupNames label

G:=SmallGroup(288,632);
// by ID

G=gap.SmallGroup(288,632);
# by ID

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

G:=Group<a,b,c,d|a^6=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations

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