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## G = C22×C6.D6order 288 = 25·32

### Direct product of C22 and C6.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C6.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C6.D6 — C2×C6.D6 — C22×C6.D6
 Lower central C32 — C22×C6.D6
 Upper central C1 — C23

Generators and relations for C22×C6.D6
G = < a,b,c,d,e | a2=b2=c6=e2=1, d6=c3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d5 >

Subgroups: 1858 in 539 conjugacy classes, 188 normal (8 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C24, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C6, C23×C4, C3×Dic3, C2×C3⋊S3, C62, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C6.D6, C6×Dic3, C22×C3⋊S3, C2×C62, S3×C22×C4, C2×C6.D6, Dic3×C2×C6, C23×C3⋊S3, C22×C6.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, S32, S3×C2×C4, S3×C23, C6.D6, C2×S32, S3×C22×C4, C2×C6.D6, C22×S32, C22×C6.D6

Smallest permutation representation of C22×C6.D6
On 48 points
Generators in S48
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)(25 35 33 31 29 27)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 40 42 44 46 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 25)(2 30)(3 35)(4 28)(5 33)(6 26)(7 31)(8 36)(9 29)(10 34)(11 27)(12 32)(13 40)(14 45)(15 38)(16 43)(17 48)(18 41)(19 46)(20 39)(21 44)(22 37)(23 42)(24 47)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 4A ··· 4P 6A ··· 6N 6O ··· 6U 12A ··· 12P order 1 2 ··· 2 2 ··· 2 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 9 ··· 9 2 2 4 3 ··· 3 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D6 D6 C4×S3 S32 C6.D6 C2×S32 kernel C22×C6.D6 C2×C6.D6 Dic3×C2×C6 C23×C3⋊S3 C22×C3⋊S3 C22×Dic3 C2×Dic3 C22×C6 C2×C6 C23 C22 C22 # reps 1 12 2 1 16 2 12 2 16 1 4 3

Matrix representation of C22×C6.D6 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12

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

C22×C6.D6 in GAP, Magma, Sage, TeX

C_2^2\times C_6.D_6
% in TeX

G:=Group("C2^2xC6.D6");
// GroupNames label

G:=SmallGroup(288,972);
// by ID

G=gap.SmallGroup(288,972);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^6=e^2=1,d^6=c^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

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