direct product, metabelian, supersoluble, monomial
Aliases: C3×D4⋊2S3, Dic6⋊3C6, C12.37D6, C62.13C22, (C4×S3)⋊2C6, D4⋊2(C3×S3), (C3×D4)⋊5S3, (C3×D4)⋊3C6, C3⋊D4⋊2C6, C4.5(S3×C6), (C2×C6).7D6, (S3×C12)⋊6C2, C12.5(C2×C6), D6.2(C2×C6), (C2×Dic3)⋊3C6, (C3×Dic6)⋊8C2, (C6×Dic3)⋊9C2, (D4×C32)⋊4C2, C32⋊9(C4○D4), C6.6(C22×C6), C22.1(S3×C6), (C3×C6).24C23, C6.45(C22×S3), Dic3.3(C2×C6), (S3×C6).11C22, (C3×C12).21C22, (C3×Dic3).13C22, (C2×C6).(C2×C6), C2.7(S3×C2×C6), C3⋊2(C3×C4○D4), (C3×C3⋊D4)⋊6C2, SmallGroup(144,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊2S3
G = < a,b,c,d,e | a3=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 164 in 88 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C62, D4⋊2S3, C3×C4○D4, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, C3×D4⋊2S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, D4⋊2S3, C3×C4○D4, S3×C2×C6, C3×D4⋊2S3
►On 24 points - transitive group 24T209
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(6 8)(9 11)(13 15)(18 20)(22 24)
(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)
G:=sub<Sym(24)| (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(6,8)(9,11)(13,15)(18,20)(22,24), (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)>;
G:=Group( (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(6,8)(9,11)(13,15)(18,20)(22,24), (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) );
G=PermutationGroup([[(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(2,4),(6,8),(9,11),(13,15),(18,20),(22,24)], [(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)]])
G:=TransitiveGroup(24,209);
C3×D4⋊2S3 is a maximal subgroup of
Dic6⋊3D6 Dic6.19D6 D12.22D6 Dic6.20D6 Dic6.24D6 Dic6⋊12D6 D12⋊13D6 C3×S3×C4○D4 C62.13D6 Dic18⋊2C6 C62.16D6
C3×D4⋊2S3 is a maximal quotient of
C3×D4×Dic3 C62.13D6 Dic18⋊2C6
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6I | 6J | ··· | 6O | 6P | 6Q | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 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 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C3×C4○D4 | D4⋊2S3 | C3×D4⋊2S3 |
| kernel | C3×D4⋊2S3 | C3×Dic6 | S3×C12 | C6×Dic3 | C3×C3⋊D4 | D4×C32 | D4⋊2S3 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | C3×D4 | C12 | C2×C6 | C32 | D4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 |
Matrix representation of C3×D4⋊2S3 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 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 |
| 0 | 0 | 4 | 6 |
| 2 | 3 | 1 | 0 |
| 6 | 1 | 2 | 2 |
| 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))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[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],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1] >;
C3×D4⋊2S3 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_2S_3
% in TeX
G:=Group("C3xD4:2S3"); // GroupNames label
G:=SmallGroup(144,163);
// by ID
G=gap.SmallGroup(144,163);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,260,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^3=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,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations