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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 [×6], C2 [×4], C3 [×2], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6 [×2], C6 [×12], C6 [×23], C2×C4 [×8], C23, C23 [×6], C23 [×4], C32, Dic3 [×4], C12 [×4], C2×C6 [×2], C2×C6 [×20], C2×C6 [×59], C22⋊C4 [×4], C22×C4 [×2], C24, C3×C6, C3×C6 [×6], C3×C6 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×8], C22×C6 [×2], C22×C6 [×12], C22×C6 [×23], C2×C22⋊C4, C3×Dic3 [×4], C62, C62 [×10], C62 [×12], C6.D4 [×4], C3×C22⋊C4 [×4], C22×Dic3 [×2], C22×C12 [×2], C23×C6 [×2], C23×C6, C6×Dic3 [×4], C6×Dic3 [×4], C2×C62, C2×C62 [×6], C2×C62 [×4], C2×C6.D4, C6×C22⋊C4, C3×C6.D4 [×4], Dic3×C2×C6 [×2], C22×C62, C6×C6.D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×4], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3×S3, C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×6], C3×D4 [×4], C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3 [×4], S3×C6 [×3], C6.D4 [×4], C3×C22⋊C4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C22×C12, C6×D4 [×2], C6×Dic3 [×6], C3×C3⋊D4 [×4], S3×C2×C6, C2×C6.D4, C6×C22⋊C4, C3×C6.D4 [×4], Dic3×C2×C6, C6×C3⋊D4 [×2], C6×C6.D4

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

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

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

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

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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