direct product, metabelian, supersoluble, monomial
Aliases: C3×D6.3D6, C62.87D6, D6.3(S3×C6), C3⋊D12⋊2C6, (C6×Dic3)⋊6C6, (C6×Dic3)⋊9S3, (S3×Dic3)⋊5C6, (S3×C6).23D6, C6.D6⋊2C6, C32⋊2Q8⋊4C6, C32⋊7D4⋊5C6, C33⋊21(C4○D4), C62.23(C2×C6), Dic3.3(S3×C6), (C3×Dic3).47D6, C32⋊24(C4○D12), (C3×C62).17C22, (C32×C6).30C23, C32⋊23(D4⋊2S3), (C32×Dic3).27C22, C2.12(S32×C6), (C2×C6).13S32, C6.11(S3×C2×C6), C6.114(C2×S32), C3⋊4(C3×C4○D12), (C3×C3⋊D4)⋊7S3, C3⋊D4⋊3(C3×S3), (C3×C3⋊D4)⋊1C6, C22.1(C3×S32), (Dic3×C3×C6)⋊9C2, C3⋊3(C3×D4⋊2S3), (S3×C6).3(C2×C6), (C2×C6).14(S3×C6), (C3×S3×Dic3)⋊12C2, C32⋊8(C3×C4○D4), (C2×Dic3)⋊3(C3×S3), (C3×C6.D6)⋊5C2, (C32×C3⋊D4)⋊1C2, (C3×C32⋊7D4)⋊7C2, (S3×C3×C6).12C22, (C3×C3⋊D12)⋊14C2, (C3×C32⋊2Q8)⋊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
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, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, C4○D12, D4⋊2S3, C3×C4○D4, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C6×Dic3, C3×C3⋊D4, C3×C3⋊D4, C32⋊7D4, C6×C12, D4×C32, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C6×C3⋊S3, C3×C62, D6.3D6, C3×C4○D12, C3×D4⋊2S3, C3×S3×Dic3, C3×C6.D6, C3×C3⋊D12, C3×C32⋊2Q8, Dic3×C3×C6, C32×C3⋊D4, C3×C32⋊7D4, C3×D6.3D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, C4○D12, D4⋊2S3, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D6.3D6, C3×C4○D12, C3×D4⋊2S3, S32×C6, C3×D6.3D6
►On 24 points - transitive group 24T1281
(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