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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 [×6], C2 [×8], C3 [×2], C3, C4 [×8], C22 [×7], C22 [×28], S3 [×24], C6 [×14], C6 [×7], C2×C4 [×28], C23, C23 [×14], C32, Dic3 [×8], C12 [×8], D6 [×84], C2×C6 [×14], C2×C6 [×7], C22×C4 [×14], C24, C3⋊S3 [×8], C3×C6, C3×C6 [×6], C4×S3 [×32], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×42], C22×C6 [×2], C22×C6, C23×C4, C3×Dic3 [×8], C2×C3⋊S3 [×28], C62 [×7], S3×C2×C4 [×24], C22×Dic3 [×2], C22×C12 [×2], S3×C23 [×3], C6.D6 [×16], C6×Dic3 [×12], C22×C3⋊S3 [×14], C2×C62, S3×C22×C4 [×2], C2×C6.D6 [×12], Dic3×C2×C6 [×2], C23×C3⋊S3, C22×C6.D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3 [×2], C2×C4 [×28], C23 [×15], D6 [×14], C22×C4 [×14], C24, C4×S3 [×8], C22×S3 [×14], C23×C4, S32, S3×C2×C4 [×12], S3×C23 [×2], C6.D6 [×4], C2×S32 [×3], S3×C22×C4 [×2], C2×C6.D6 [×6], C22×S32, C22×C6.D6

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

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

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

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

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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