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## G = C6×C6.D4order 288 = 25·32

### Direct product of C6 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C2×C6 — C6×C6.D4
 Lower central C3 — C6 — C6×C6.D4
 Upper central C1 — C22×C6 — C23×C6

Generators and relations for C6×C6.D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 618 in 331 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23×C6, C6×Dic3, C6×Dic3, C2×C62, C2×C62, C2×C62, C2×C6.D4, C6×C22⋊C4, C3×C6.D4, Dic3×C2×C6, C22×C62, C6×C6.D4
Quotients:

Smallest permutation representation of C6×C6.D4
On 48 points
Generators in S48
(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 21 5 19 3 23)(2 22 6 20 4 24)(7 29 11 27 9 25)(8 30 12 28 10 26)(13 39 15 41 17 37)(14 40 16 42 18 38)(31 43 33 45 35 47)(32 44 34 46 36 48)
(1 18 26 43)(2 13 27 44)(3 14 28 45)(4 15 29 46)(5 16 30 47)(6 17 25 48)(7 36 20 41)(8 31 21 42)(9 32 22 37)(10 33 23 38)(11 34 24 39)(12 35 19 40)
(1 40 19 18)(2 41 20 13)(3 42 21 14)(4 37 22 15)(5 38 23 16)(6 39 24 17)(7 44 27 36)(8 45 28 31)(9 46 29 32)(10 47 30 33)(11 48 25 34)(12 43 26 35)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 3C 3D 3E 4A ··· 4H 6A ··· 6N 6O ··· 6BO 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 1 1 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 S3×C6 C3×C3⋊D4 kernel C6×C6.D4 C3×C6.D4 Dic3×C2×C6 C22×C62 C2×C6.D4 C2×C62 C6.D4 C22×Dic3 C23×C6 C22×C6 C23×C6 C62 C22×C6 C22×C6 C24 C2×C6 C2×C6 C23 C23 C22 # reps 1 4 2 1 2 8 8 4 2 16 1 4 4 3 2 8 8 8 6 16

Matrix representation of C6×C6.D4 in GL4(𝔽13) generated by

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

C6×C6.D4 in GAP, Magma, Sage, TeX

C_6\times C_6.D_4
% in TeX

G:=Group("C6xC6.D4");
// GroupNames label

G:=SmallGroup(288,723);
// by ID

G=gap.SmallGroup(288,723);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,1094,9414]);
// Polycyclic

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

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