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## G = C3×C6.D6order 216 = 23·33

### Direct product of C3 and C6.D6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C6.D6
G = < a,b,c,d | a3=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 268 in 90 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C3 [×4], C4 [×2], C22, S3 [×6], C6, C6 [×2], C6 [×6], C2×C4, C32, C32 [×2], C32 [×4], Dic3 [×2], C12 [×6], D6 [×3], C2×C6, C3×S3 [×6], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], C4×S3 [×2], C2×C12, C33, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×3], C2×C3⋊S3, C3×C3⋊S3 [×2], C32×C6, C6.D6, S3×C12 [×2], C32×Dic3 [×2], C6×C3⋊S3, C3×C6.D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3 [×2], C6 [×3], C2×C4, C12 [×2], D6 [×2], C2×C6, C3×S3 [×2], C4×S3 [×2], C2×C12, S32, S3×C6 [×2], C6.D6, S3×C12 [×2], C3×S32, C3×C6.D6

Permutation representations of C3×C6.D6
On 24 points - transitive group 24T544
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)
(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)(2 15)(3 20)(4 13)(5 18)(6 23)(7 16)(8 21)(9 14)(10 19)(11 24)(12 17)

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

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

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

G:=TransitiveGroup(24,544);

C3×C6.D6 is a maximal subgroup of
C3⋊S3.2D12  C33⋊C4⋊C4  Dic36S32  C3⋊S34D12  C335(C2×Q8)  D6.S32  Dic3.S32  S32×C12
C3×C6.D6 is a maximal quotient of
C3×Dic32

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 12A ··· 12H 12I ··· 12T order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 9 9 1 1 2 ··· 2 4 4 4 3 3 3 3 1 1 2 ··· 2 4 4 4 9 9 9 9 3 ··· 3 6 ··· 6

54 irreducible representations

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

Matrix representation of C3×C6.D6 in GL4(𝔽7) generated by

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

C3×C6.D6 in GAP, Magma, Sage, TeX

C_3\times C_6.D_6
% in TeX

G:=Group("C3xC6.D6");
// GroupNames label

G:=SmallGroup(216,120);
// by ID

G=gap.SmallGroup(216,120);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,79,730,5189]);
// Polycyclic

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

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