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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 — S3×C6 — C2×S32 — C22×S32 — C2×Dic3⋊D6
 Lower central C32 — C3×C6 — C2×Dic3⋊D6
 Upper central C1 — C22 — C23

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

Subgroups: 2370 in 539 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×36], S3 [×24], C6 [×6], C6 [×15], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, Dic3 [×4], C12 [×4], D6 [×4], D6 [×84], C2×C6 [×6], C2×C6 [×19], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3⋊S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×8], D12 [×8], C2×Dic3 [×2], C3⋊D4 [×8], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×50], C22×C6 [×2], C22×C6 [×3], C22×D4, C3×Dic3 [×4], S32 [×8], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3 [×8], C2×C3⋊S3 [×10], C62, C62 [×2], C62 [×2], S3×C2×C4 [×2], C2×D12 [×2], S3×D4 [×16], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], S3×C23 [×5], C6.D6 [×4], C3⋊D12 [×8], C6×Dic3 [×2], C3×C3⋊D4 [×8], C2×S32 [×4], C2×S32 [×4], S3×C2×C6 [×2], C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C22×C3⋊S3 [×4], C2×C62, C2×S3×D4 [×2], C2×C6.D6, C2×C3⋊D12 [×2], Dic3⋊D6 [×8], C6×C3⋊D4 [×2], C22×S32, C23×C3⋊S3, C2×Dic3⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, C22×S3 [×14], C22×D4, S32, S3×D4 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×D4 [×2], Dic3⋊D6 [×2], C22×S32, C2×Dic3⋊D6

Permutation representations of C2×Dic3⋊D6
On 24 points - transitive group 24T671
Generators in S24
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 7 5 11 3 9)(2 8 6 12 4 10)(13 20 15 22 17 24)(14 21 16 23 18 19)
(1 5)(2 4)(8 12)(9 11)(13 15)(16 18)(19 21)(22 24)

G:=sub<Sym(24)| (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,20,15,22,17,24)(14,21,16,23,18,19), (1,5)(2,4)(8,12)(9,11)(13,15)(16,18)(19,21)(22,24)>;

G:=Group( (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,14,10,17)(8,13,11,16)(9,18,12,15), (1,7,5,11,3,9)(2,8,6,12,4,10)(13,20,15,22,17,24)(14,21,16,23,18,19), (1,5)(2,4)(8,12)(9,11)(13,15)(16,18)(19,21)(22,24) );

G=PermutationGroup([(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,14,10,17),(8,13,11,16),(9,18,12,15)], [(1,7,5,11,3,9),(2,8,6,12,4,10),(13,20,15,22,17,24),(14,21,16,23,18,19)], [(1,5),(2,4),(8,12),(9,11),(13,15),(16,18),(19,21),(22,24)])

G:=TransitiveGroup(24,671);

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G ··· 6Q 6R 6S 6T 6U 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 2 2 6 6 6 6 9 9 9 9 18 18 2 2 4 6 6 6 6 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 S32 S3×D4 C2×S32 Dic3⋊D6 kernel C2×Dic3⋊D6 C2×C6.D6 C2×C3⋊D12 Dic3⋊D6 C6×C3⋊D4 C22×S32 C23×C3⋊S3 C2×C3⋊D4 C2×C3⋊S3 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C23 C6 C22 C2 # reps 1 1 2 8 2 1 1 2 4 2 8 2 2 1 4 3 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
,
 12 1 0 0 0 0 12 0 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 5 0 0 0 0 10 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 3 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,3,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×Dic3⋊D6 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,977);
// by ID

G=gap.SmallGroup(288,977);
# by ID

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

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

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