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## G = C2×Dic3.D6order 288 = 25·32

### Direct product of C2 and Dic3.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×Dic3.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C6.D6 — C2×C6.D6 — C2×Dic3.D6
 Lower central C32 — C3×C6 — C2×Dic3.D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×Dic3.D6
G = < a,b,c,d,e | a2=b6=1, c2=d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=b3d5 >

Subgroups: 1106 in 339 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C2×Q8, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×Q8, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C2×Dic6, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, C6.D6, C322Q8, C3×Dic6, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×S3×Q8, Dic3.D6, C2×C6.D6, C2×C322Q8, C6×Dic6, C2×C4×C3⋊S3, C2×Dic3.D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, S32, S3×Q8, S3×C23, C2×S32, C2×S3×Q8, Dic3.D6, C22×S32, C2×Dic3.D6

Smallest permutation representation of C2×Dic3.D6
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(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 12 10 8 6 4)(13 23 21 19 17 15)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 39 41 43 45 47)(38 40 42 44 46 48)
(1 26 7 32)(2 33 8 27)(3 28 9 34)(4 35 10 29)(5 30 11 36)(6 25 12 31)(13 38 19 44)(14 45 20 39)(15 40 21 46)(16 47 22 41)(17 42 23 48)(18 37 24 43)
(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 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 37 19 43)(14 48 20 42)(15 47 21 41)(16 46 22 40)(17 45 23 39)(18 44 24 38)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4J 4K 4L 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 4 2 2 6 ··· 6 18 18 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + + - + + image C1 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 S32 S3×Q8 C2×S32 C2×S32 Dic3.D6 kernel C2×Dic3.D6 Dic3.D6 C2×C6.D6 C2×C32⋊2Q8 C6×Dic6 C2×C4×C3⋊S3 C2×Dic6 C2×C3⋊S3 Dic6 C2×Dic3 C2×C12 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 2 1 2 4 8 4 2 1 4 2 1 4

Matrix representation of C2×Dic3.D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 1 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 3 0 0 0 0 8 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 5 11 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C2×Dic3.D6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3.D_6
% in TeX

G:=Group("C2xDic3.D6");
// GroupNames label

G:=SmallGroup(288,947);
// by ID

G=gap.SmallGroup(288,947);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,675,346,80,1356,9414]);
// Polycyclic

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

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