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## G = C3×D6.3D6order 432 = 24·33

### Direct product of C3 and D6.3D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D6.3D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3×D6.3D6
 Lower central C32 — C3×C6 — C3×D6.3D6
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×D6.3D6
G = < a,b,c,d,e | a3=b6=c2=d6=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >

Subgroups: 712 in 218 conjugacy classes, 64 normal (all characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4 [×4], C22, C22 [×2], S3 [×4], C6 [×3], C6 [×17], C2×C4 [×3], D4 [×3], Q8, C32 [×3], C32 [×4], Dic3 [×3], Dic3 [×3], C12 [×10], D6, D6 [×3], C2×C6 [×3], C2×C6 [×8], C4○D4, C3×S3 [×6], C3⋊S3, C3×C6 [×3], C3×C6 [×14], Dic6 [×2], C4×S3 [×3], D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4 [×4], C2×C12 [×5], C3×D4 [×5], C3×Q8, C33, C3×Dic3 [×6], C3×Dic3 [×6], C3⋊Dic3, C3×C12 [×3], S3×C6 [×2], S3×C6 [×4], C2×C3⋊S3, C62 [×3], C62 [×5], C4○D12, D42S3, C3×C4○D4, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C3×Dic6 [×2], S3×C12 [×3], C3×D12, C6×Dic3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×2], C3×C3⋊D4 [×5], C327D4, C6×C12, D4×C32, C32×Dic3 [×3], C3×C3⋊Dic3, S3×C3×C6, C6×C3⋊S3, C3×C62, D6.3D6, C3×C4○D12, C3×D42S3, C3×S3×Dic3, C3×C6.D6, C3×C3⋊D12, C3×C322Q8, Dic3×C3×C6, C32×C3⋊D4, C3×C327D4, C3×D6.3D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3 [×2], C6 [×7], C23, D6 [×6], C2×C6 [×7], C4○D4, C3×S3 [×2], C22×S3 [×2], C22×C6, S32, S3×C6 [×6], C4○D12, D42S3, C3×C4○D4, C2×S32, S3×C2×C6 [×2], C3×S32, D6.3D6, C3×C4○D12, C3×D42S3, S32×C6, C3×D6.3D6

Permutation representations of C3×D6.3D6
On 24 points - transitive group 24T1281
Generators in S24
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 23 16 20)(14 24 17 21)(15 19 18 22)

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

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

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

G:=TransitiveGroup(24,1281);

81 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 6A 6B 6C ··· 6P 6Q ··· 6AB 6AC 6AD 6AE 6AF 6AG 6AH 6AI 12A 12B 12C 12D 12E ··· 12T 12U 12V 12W 12X 12Y order 1 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 size 1 1 2 6 18 1 1 2 ··· 2 4 4 4 3 3 6 6 18 1 1 2 ··· 2 4 ··· 4 6 6 12 12 12 18 18 3 3 3 3 6 ··· 6 12 12 12 18 18

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 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 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 S3 D6 D6 D6 C4○D4 C3×S3 C3×S3 S3×C6 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 S32 D4⋊2S3 C2×S32 C3×S32 D6.3D6 C3×D4⋊2S3 S32×C6 C3×D6.3D6 kernel C3×D6.3D6 C3×S3×Dic3 C3×C6.D6 C3×C3⋊D12 C3×C32⋊2Q8 Dic3×C3×C6 C32×C3⋊D4 C3×C32⋊7D4 D6.3D6 S3×Dic3 C6.D6 C3⋊D12 C32⋊2Q8 C6×Dic3 C3×C3⋊D4 C32⋊7D4 C6×Dic3 C3×C3⋊D4 C3×Dic3 S3×C6 C62 C33 C2×Dic3 C3⋊D4 Dic3 D6 C2×C6 C32 C32 C3 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 3 1 2 2 2 2 6 2 4 4 4 8 1 1 1 2 2 2 2 4

Matrix representation of C3×D6.3D6 in GL4(𝔽7) generated by

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

C3×D6.3D6 in GAP, Magma, Sage, TeX

C_3\times D_6._3D_6
% in TeX

G:=Group("C3xD6.3D6");
// GroupNames label

G:=SmallGroup(432,652);
// by ID

G=gap.SmallGroup(432,652);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,2028,14118]);
// Polycyclic

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

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