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## G = C3×Dic3⋊D6order 432 = 24·33

### Direct product of C3 and Dic3⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×Dic3⋊D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — S32×C6 — C3×Dic3⋊D6
 Lower central C32 — C3×C6 — C3×Dic3⋊D6
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×Dic3⋊D6
G = < a,b,c,d,e | a3=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: 1152 in 290 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×12], C6, C6 [×2], C6 [×22], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×2], C12 [×6], D6 [×2], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×16], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6 [×2], C3×C6 [×15], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C3⋊D4 [×2], C2×C12, C3×D4 [×8], C22×S3 [×5], C22×C6 [×2], C33, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], S32 [×2], S3×C6 [×4], S3×C6 [×20], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×6], S3×D4 [×2], C6×D4, S3×C32 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3, C32×C6, C32×C6, C6.D6, C3⋊D12 [×2], S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×4], C3×C3⋊D4 [×4], D4×C32 [×2], C2×S32, S3×C2×C6 [×5], C22×C3⋊S3, C32×Dic3 [×2], C3×S32 [×2], S3×C3×C6 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C3×C62, Dic3⋊D6, C3×S3×D4 [×2], C3×C6.D6, C3×C3⋊D12 [×2], C32×C3⋊D4 [×2], S32×C6, C2×C6×C3⋊S3, C3×Dic3⋊D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3 [×2], C6 [×7], D4 [×2], C23, D6 [×6], C2×C6 [×7], C2×D4, C3×S3 [×2], C3×D4 [×2], C22×S3 [×2], C22×C6, S32, S3×C6 [×6], S3×D4 [×2], C6×D4, C2×S32, S3×C2×C6 [×2], C3×S32, Dic3⋊D6, C3×S3×D4 [×2], S32×C6, C3×Dic3⋊D6

Permutation representations of C3×Dic3⋊D6
On 24 points - transitive group 24T1282
Generators in S24
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(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 10 4 7)(2 9 5 12)(3 8 6 11)(13 23 16 20)(14 22 17 19)(15 21 18 24)
(1 5 3)(2 6 4)(7 12 11 10 9 8)(13 18 17 16 15 14)(19 23 21)(20 24 22)
(1 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)

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

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

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

G:=TransitiveGroup(24,1282);

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 6A 6B 6C ··· 6J 6K ··· 6Y 6Z 6AA 6AB 6AC 6AD 6AE 6AF 6AG 6AH ··· 6AM 6AN 6AO 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 12 12 12 12 12 ··· 12 size 1 1 2 6 6 9 9 18 1 1 2 ··· 2 4 4 4 6 6 1 1 2 ··· 2 4 ··· 4 6 6 6 6 9 9 9 9 12 ··· 12 18 18 6 6 6 6 12 ··· 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D6 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S3×C6 S32 S3×D4 C2×S32 C3×S32 Dic3⋊D6 C3×S3×D4 S32×C6 C3×Dic3⋊D6 kernel C3×Dic3⋊D6 C3×C6.D6 C3×C3⋊D12 C32×C3⋊D4 S32×C6 C2×C6×C3⋊S3 Dic3⋊D6 C6.D6 C3⋊D12 C3×C3⋊D4 C2×S32 C22×C3⋊S3 C3×C3⋊D4 C3×C3⋊S3 C3×Dic3 S3×C6 C62 C3⋊D4 C3⋊S3 Dic3 D6 C2×C6 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 2 2 2 2 2 4 4 4 4 4 1 2 1 2 2 4 2 4

Matrix representation of C3×Dic3⋊D6 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 6 5 6 4 4 1 3 1 1 3 5 1 6 3 6
,
 4 0 5 0 4 6 4 6 5 0 3 0 4 2 0 1
,
 5 4 6 1 0 4 5 6 3 0 3 4 6 6 6 2
,
 6 3 1 5 2 2 2 0 6 6 4 2 6 1 1 2
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,4,1,1,6,4,1,6,5,1,3,3,6,3,5,6],[4,4,5,4,0,6,0,2,5,4,3,0,0,6,0,1],[5,0,3,6,4,4,0,6,6,5,3,6,1,6,4,2],[6,2,6,6,3,2,6,1,1,2,4,1,5,0,2,2] >;

C3×Dic3⋊D6 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\rtimes D_6
% in TeX

G:=Group("C3xDic3:D6");
// GroupNames label

G:=SmallGroup(432,659);
// by ID

G=gap.SmallGroup(432,659);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,303,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=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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