direct product, metabelian, supersoluble, monomial
Aliases: C3×S3×D4, C12⋊5D6, D12⋊3C6, C62⋊3C22, C12⋊(C2×C6), C4⋊1(S3×C6), (C2×C6)⋊7D6, C3⋊2(C6×D4), (C4×S3)⋊1C6, (C3×D4)⋊2C6, D6⋊2(C2×C6), C3⋊D4⋊1C6, (S3×C12)⋊5C2, (C3×D12)⋊8C2, C22⋊3(S3×C6), C32⋊10(C2×D4), (S3×C6)⋊8C22, (C22×S3)⋊3C6, (C3×C12)⋊3C22, Dic3⋊1(C2×C6), (D4×C32)⋊3C2, C6.5(C22×C6), C6.44(C22×S3), (C3×C6).23C23, (C3×Dic3)⋊8C22, (S3×C2×C6)⋊5C2, C2.6(S3×C2×C6), (C2×C6)⋊2(C2×C6), (C3×C3⋊D4)⋊5C2, SmallGroup(144,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3×D4
G = < a,b,c,d,e | a3=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 260 in 116 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, D4, C23, C32, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C3×Dic3, C3×C12, S3×C6, S3×C6, S3×C6, C62, S3×D4, C6×D4, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, S3×C2×C6, C3×S3×D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, S3×C6, S3×D4, C6×D4, S3×C2×C6, C3×S3×D4
►On 24 points - transitive group 24T208
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
G:=sub<Sym(24)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)>;
G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24) );
G=PermutationGroup([[(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24)]])
G:=TransitiveGroup(24,208);
C3×S3×D4 is a maximal subgroup of
D12⋊9D6 D12.7D6 D12⋊12D6 D12⋊13D6
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | ··· | 6I | 6J | 6K | 6L | 6M | 6N | ··· | 6S | 6T | 6U | 6V | 6W | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 |
45 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C3×S3 | C3×D4 | S3×C6 | S3×C6 | S3×D4 | C3×S3×D4 |
| kernel | C3×S3×D4 | S3×C12 | C3×D12 | C3×C3⋊D4 | D4×C32 | S3×C2×C6 | S3×D4 | C4×S3 | D12 | C3⋊D4 | C3×D4 | C22×S3 | C3×D4 | C3×S3 | C12 | C2×C6 | D4 | S3 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 1 | 2 |
Matrix representation of C3×S3×D4 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 6 | 0 | 3 | 1 |
| 5 | 3 | 6 | 0 |
| 1 | 6 | 4 | 5 |
| 3 | 3 | 2 | 6 |
| 6 | 6 | 1 | 1 |
| 0 | 6 | 0 | 1 |
| 0 | 5 | 1 | 1 |
| 0 | 0 | 0 | 1 |
| 3 | 1 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 4 | 5 | 6 | 3 |
| 0 | 3 | 4 | 1 |
| 3 | 2 | 6 | 3 |
| 6 | 0 | 4 | 5 |
| 6 | 6 | 5 | 2 |
| 0 | 0 | 0 | 6 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[6,0,0,0,6,6,5,0,1,0,1,0,1,1,1,1],[3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[3,6,6,0,2,0,6,0,6,4,5,0,3,5,2,6] >;
C3×S3×D4 in GAP, Magma, Sage, TeX
C_3\times S_3\times D_4
% in TeX
G:=Group("C3xS3xD4"); // GroupNames label
G:=SmallGroup(144,162);
// by ID
G=gap.SmallGroup(144,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,260,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^4=e^2=1,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,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations