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G = C3×D6.3D6order 432 = 24·33

Direct product of C3 and D6.3D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×D6.3D6, C62.87D6, D6.3(S3×C6), C3⋊D122C6, (C6×Dic3)⋊6C6, (C6×Dic3)⋊9S3, (S3×Dic3)⋊5C6, (S3×C6).23D6, C6.D62C6, C322Q84C6, C327D45C6, C3321(C4○D4), C62.23(C2×C6), Dic3.3(S3×C6), (C3×Dic3).47D6, C3224(C4○D12), (C3×C62).17C22, (C32×C6).30C23, C3223(D42S3), (C32×Dic3).27C22, C2.12(S32×C6), (C2×C6).13S32, C6.11(S3×C2×C6), C6.114(C2×S32), C34(C3×C4○D12), (C3×C3⋊D4)⋊7S3, C3⋊D43(C3×S3), (C3×C3⋊D4)⋊1C6, C22.1(C3×S32), (Dic3×C3×C6)⋊9C2, C33(C3×D42S3), (S3×C6).3(C2×C6), (C2×C6).14(S3×C6), (C3×S3×Dic3)⋊12C2, C328(C3×C4○D4), (C2×Dic3)⋊3(C3×S3), (C3×C6.D6)⋊5C2, (C32×C3⋊D4)⋊1C2, (C3×C327D4)⋊7C2, (S3×C3×C6).12C22, (C3×C3⋊D12)⋊14C2, (C3×C322Q8)⋊10C2, (C6×C3⋊S3).25C22, C3⋊Dic3.10(C2×C6), (C3×C6).21(C22×C6), (C3×Dic3).4(C2×C6), (C3×C6).135(C22×S3), (C3×C3⋊Dic3).36C22, (C2×C3⋊S3).8(C2×C6), SmallGroup(432,652)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×D6.3D6
C1C3C32C3×C6C32×C6S3×C3×C6C3×S3×Dic3 — C3×D6.3D6
C32C3×C6 — C3×D6.3D6
C1C6C2×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 2A2B2C2D3A3B3C···3H3I3J3K4A4B4C4D4E6A6B6C···6P6Q···6AB6AC6AD6AE6AF6AG6AH6AI12A12B12C12D12E···12T12U12V12W12X12Y
order12222333···333344444666···66···666666661212121212···121212121212
size112618112···2444336618112···24···466121212181833336···61212121818

81 irreducible representations

dim11111111111111112222222222222244444444
type++++++++++++++-+
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3S3D6D6D6C4○D4C3×S3C3×S3S3×C6S3×C6S3×C6C4○D12C3×C4○D4C3×C4○D12S32D42S3C2×S32C3×S32D6.3D6C3×D42S3S32×C6C3×D6.3D6
kernelC3×D6.3D6C3×S3×Dic3C3×C6.D6C3×C3⋊D12C3×C322Q8Dic3×C3×C6C32×C3⋊D4C3×C327D4D6.3D6S3×Dic3C6.D6C3⋊D12C322Q8C6×Dic3C3×C3⋊D4C327D4C6×Dic3C3×C3⋊D4C3×Dic3S3×C6C62C33C2×Dic3C3⋊D4Dic3D6C2×C6C32C32C3C2×C6C32C6C22C3C3C2C1
# reps11111111222222221131222262444811122224

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

4000
0400
0040
0004
,
0031
5460
1655
3320
,
3501
6445
2104
4010
,
4144
5653
3102
5244
,
4546
6060
3331
3420
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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