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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 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×10], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×17], Q8 [×16], C23, C32, Dic3 [×8], Dic3 [×6], C12 [×4], C12 [×10], D6 [×18], C2×C6 [×2], C2×C6, C22×C4 [×3], C2×Q8 [×12], C3⋊S3 [×4], C3×C6, C3×C6 [×2], Dic6 [×8], Dic6 [×16], C4×S3 [×28], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×8], C22×S3 [×3], C22×Q8, C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C2×Dic6 [×2], C2×Dic6 [×4], S3×C2×C4 [×7], S3×Q8 [×16], C6×Q8 [×2], C6.D6 [×8], C322Q8 [×8], C3×Dic6 [×8], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×S3×Q8 [×2], Dic3.D6 [×8], C2×C6.D6 [×2], C2×C322Q8 [×2], C6×Dic6 [×2], C2×C4×C3⋊S3, C2×Dic3.D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], Q8 [×4], C23 [×15], D6 [×14], C2×Q8 [×6], C24, C22×S3 [×14], C22×Q8, S32, S3×Q8 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×Q8 [×2], Dic3.D6 [×2], C22×S32, C2×Dic3.D6

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

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

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

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

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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