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G = C3×C6.D6order 216 = 23·33

Direct product of C3 and C6.D6

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3×C6.D6, C6.26S32, C3⋊S32C12, C31(S3×C12), C6.2(S3×C6), C335(C2×C4), C329(C4×S3), (C3×C6).39D6, C325(C2×C12), (C3×Dic3)⋊5S3, Dic32(C3×S3), (C3×Dic3)⋊3C6, (C32×Dic3)⋊4C2, (C32×C6).2C22, C2.2(C3×S32), (C3×C3⋊S3)⋊1C4, (C2×C3⋊S3).2C6, (C6×C3⋊S3).1C2, (C3×C6).7(C2×C6), (C3×Dic3)(C3×Dic3), SmallGroup(216,120)

Series: Derived Chief Lower central Upper central

C1C32 — C3×C6.D6
C1C3C32C3×C6C32×C6C32×Dic3 — C3×C6.D6
C32 — C3×C6.D6
C1C6

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 2A2B2C3A3B3C···3H3I3J3K4A4B4C4D6A6B6C···6H6I6J6K6L6M6N6O12A···12H12I···12T
order1222333···33334444666···6666666612···1212···12
size1199112···24443333112···244499993···36···6

54 irreducible representations

dim111111112222224444
type+++++++
imageC1C2C2C3C4C6C6C12S3D6C3×S3C4×S3S3×C6S3×C12S32C6.D6C3×S32C3×C6.D6
kernelC3×C6.D6C32×Dic3C6×C3⋊S3C6.D6C3×C3⋊S3C3×Dic3C2×C3⋊S3C3⋊S3C3×Dic3C3×C6Dic3C32C6C3C6C3C2C1
# reps121244282244481122

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

4000
0400
0040
0004
,
1046
2410
6132
4451
,
3022
3316
2311
3140
,
2402
4223
1645
1636
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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