direct product, metabelian, supersoluble, monomial
Aliases: C3×D4⋊6D6, C32⋊72+ 1+4, C62.149C23, (S3×D4)⋊4C6, D4⋊6(S3×C6), (C6×D4)⋊7C6, C4○D12⋊5C6, (C3×D4)⋊24D6, (C6×D4)⋊16S3, D12⋊8(C2×C6), (C2×C12)⋊17D6, C23⋊3(S3×C6), (C22×C6)⋊5D6, D4⋊2S3⋊4C6, Dic6⋊8(C2×C6), C6.7(C23×C6), (C6×C12)⋊10C22, (C3×C6).44C24, C6.75(S3×C23), D6.3(C22×C6), (S3×C12)⋊12C22, (C3×D12)⋊34C22, (S3×C6).30C23, C12.21(C22×C6), (C2×C62)⋊10C22, C3⋊1(C3×2+ 1+4), C12.172(C22×S3), (C3×C12).122C23, (C6×Dic3)⋊20C22, (C3×Dic6)⋊33C22, (D4×C32)⋊20C22, Dic3.4(C22×C6), (C3×Dic3).31C23, (C2×C4)⋊3(S3×C6), (D4×C3×C6)⋊11C2, (C3×S3×D4)⋊11C2, C4.21(S3×C2×C6), (C4×S3)⋊1(C2×C6), (C2×C12)⋊3(C2×C6), (C3×D4)⋊7(C2×C6), (C2×D4)⋊7(C3×S3), C3⋊D4⋊3(C2×C6), C2.8(S3×C22×C6), C22.6(S3×C2×C6), (C2×C3⋊D4)⋊11C6, (C6×C3⋊D4)⋊25C2, (S3×C2×C6)⋊14C22, (C22×C6)⋊6(C2×C6), (C3×C4○D12)⋊15C2, (C22×S3)⋊3(C2×C6), (C2×Dic3)⋊4(C2×C6), (C2×C6).2(C22×C6), (C3×D4⋊2S3)⋊11C2, (C3×C3⋊D4)⋊20C22, (C2×C6).21(C22×S3), SmallGroup(288,994)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊6D6
G = < a,b,c,d,e | a3=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 810 in 359 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, 2+ 1+4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, C6×D4, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, D4⋊6D6, C3×2+ 1+4, C3×C4○D12, C3×S3×D4, C3×D4⋊2S3, C6×C3⋊D4, D4×C3×C6, C3×D4⋊6D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, 2+ 1+4, S3×C6, S3×C23, C23×C6, S3×C2×C6, D4⋊6D6, C3×2+ 1+4, S3×C22×C6, C3×D4⋊6D6
►On 24 points - transitive group 24T588
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 4 15 18)(2 13 16 5)(3 6 17 14)(7 10 19 22)(8 23 20 11)(9 12 21 24)
(2 16)(4 18)(6 14)(7 19)(9 21)(11 23)
(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 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)
G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,4,15,18)(2,13,16,5)(3,6,17,14)(7,10,19,22)(8,23,20,11)(9,12,21,24), (2,16)(4,18)(6,14)(7,19)(9,21)(11,23), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,21)(2,20)(3,19)(4,24)(5,23)(6,22)(7,17)(8,16)(9,15)(10,14)(11,13)(12,18)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,4,15,18)(2,13,16,5)(3,6,17,14)(7,10,19,22)(8,23,20,11)(9,12,21,24), (2,16)(4,18)(6,14)(7,19)(9,21)(11,23), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,21)(2,20)(3,19)(4,24)(5,23)(6,22)(7,17)(8,16)(9,15)(10,14)(11,13)(12,18) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,4,15,18),(2,13,16,5),(3,6,17,14),(7,10,19,22),(8,23,20,11),(9,12,21,24)], [(2,16),(4,18),(6,14),(7,19),(9,21),(11,23)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,21),(2,20),(3,19),(4,24),(5,23),(6,22),(7,17),(8,16),(9,15),(10,14),(11,13),(12,18)]])
G:=TransitiveGroup(24,588);
81 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6U | 6V | ··· | 6AG | 6AH | ··· | 6AO | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | D6 | C3×S3 | S3×C6 | S3×C6 | S3×C6 | 2+ 1+4 | D4⋊6D6 | C3×2+ 1+4 | C3×D4⋊6D6 |
| kernel | C3×D4⋊6D6 | C3×C4○D12 | C3×S3×D4 | C3×D4⋊2S3 | C6×C3⋊D4 | D4×C3×C6 | D4⋊6D6 | C4○D12 | S3×D4 | D4⋊2S3 | C2×C3⋊D4 | C6×D4 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C32 | C3 | C3 | C1 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 8 | 8 | 8 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×D4⋊6D6 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 6 | 4 |
| 3 | 0 | 6 | 3 |
| 4 | 4 | 2 | 6 |
| 6 | 1 | 4 | 5 |
| 6 | 1 | 6 | 0 |
| 6 | 6 | 2 | 3 |
| 6 | 5 | 3 | 3 |
| 0 | 4 | 3 | 6 |
| 4 | 4 | 1 | 5 |
| 5 | 4 | 6 | 5 |
| 6 | 4 | 6 | 3 |
| 4 | 4 | 5 | 0 |
| 3 | 3 | 5 | 0 |
| 6 | 0 | 4 | 6 |
| 6 | 1 | 3 | 2 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,4,6,0,0,4,1,6,6,2,4,4,3,6,5],[6,6,6,0,1,6,5,4,6,2,3,3,0,3,3,6],[4,5,6,4,4,4,4,4,1,6,6,5,5,5,3,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1] >;
C3×D4⋊6D6 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_6D_6
% in TeX
G:=Group("C3xD4:6D6"); // GroupNames label
G:=SmallGroup(288,994);
// by ID
G=gap.SmallGroup(288,994);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,555,1571,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations