direct product, metabelian, supersoluble, monomial
Aliases: C3×D4.S3, Dic6⋊2C6, C12.34D6, C32⋊8SD16, C3⋊C8⋊2C6, D4.(C3×S3), C4.2(S3×C6), C6.8(C3×D4), C12.2(C2×C6), (C3×D4).1C6, (C3×D4).6S3, C3⋊2(C3×SD16), (C3×C6).29D4, (C3×Dic6)⋊3C2, C6.30(C3⋊D4), (C3×C12).9C22, (D4×C32).1C2, (C3×C3⋊C8)⋊9C2, C2.5(C3×C3⋊D4), SmallGroup(144,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.S3
G = < a,b,c,d,e | a3=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Permutation representations of C3×D4.S3
►On 24 points - transitive group 24T245
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(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)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(22 24)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (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)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (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)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(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),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(22,24)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
G:=TransitiveGroup(24,245);
C3×D4.S3 is a maximal subgroup of
Dic6.19D6 Dic6⋊D6 Dic6.D6 D12.7D6 Dic6.20D6 D12.8D6 D12⋊5D6 C3×S3×SD16 He3⋊8SD16 Dic18⋊C6 He3⋊9SD16
C3×D4.S3 is a maximal quotient of
He3⋊8SD16 Dic18⋊C6
36 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6M | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 6 | 6 | 6 | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×SD16 | C3×C3⋊D4 | D4.S3 | C3×D4.S3 |
| kernel | C3×D4.S3 | C3×C3⋊C8 | C3×Dic6 | D4×C32 | D4.S3 | C3⋊C8 | Dic6 | C3×D4 | C3×D4 | C3×C6 | C12 | C32 | D4 | C6 | C6 | C4 | C3 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 |
Matrix representation of C3×D4.S3 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 1 | 6 | 4 |
| 2 | 6 | 4 | 4 |
| 4 | 4 | 4 | 6 |
| 5 | 2 | 1 | 0 |
| 3 | 6 | 6 | 0 |
| 5 | 1 | 1 | 6 |
| 3 | 3 | 3 | 1 |
| 2 | 5 | 6 | 0 |
| 3 | 1 | 4 | 5 |
| 1 | 3 | 3 | 5 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 6 | 6 |
| 4 | 5 | 6 | 5 |
| 5 | 2 | 1 | 6 |
| 5 | 5 | 2 | 4 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,2,4,5,1,6,4,2,6,4,4,1,4,4,6,0],[3,5,3,2,6,1,3,5,6,1,3,6,0,6,1,0],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[4,4,5,5,0,5,2,5,6,6,1,2,6,5,6,4] >;
C3×D4.S3 in GAP, Magma, Sage, TeX
C_3\times D_4.S_3
% in TeX
G:=Group("C3xD4.S3"); // GroupNames label
G:=SmallGroup(144,81);
// by ID
G=gap.SmallGroup(144,81);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,867,441,69,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
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