direct product, metabelian, supersoluble, monomial
Aliases: C3×C42⋊4S3, C12≀C2, D12⋊1C12, C122⋊9C2, Dic6⋊1C12, C12.92D12, C62.101D4, C32⋊8C4≀C2, (C4×C12)⋊12S3, (C4×C12)⋊12C6, C4.6(S3×C12), C42⋊7(C3×S3), (C3×D12)⋊10C4, C12.84(C4×S3), C4○D12.1C6, C4.17(C3×D12), C12.33(C3×D4), C4.Dic3⋊1C6, C12.16(C2×C12), C6.42(D6⋊C4), (C3×Dic6)⋊10C4, (C3×C12).135D4, (C2×C12).430D6, (C6×C12).311C22, C3⋊1(C3×C4≀C2), C2.3(C3×D6⋊C4), (C2×C4).69(S3×C6), (C2×C6).36(C3×D4), C6.1(C3×C22⋊C4), (C3×C12).96(C2×C4), (C3×C4○D12).3C2, C22.7(C3×C3⋊D4), (C2×C12).112(C2×C6), (C2×C6).60(C3⋊D4), (C3×C4.Dic3)⋊17C2, (C3×C6).41(C22⋊C4), SmallGroup(288,239)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊4S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, ebe=bc=cb, bd=db, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 242 in 103 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C4≀C2, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C4.Dic3, C4×C12, C4×C12, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, C42⋊4S3, C3×C4≀C2, C3×C4.Dic3, C122, C3×C4○D12, C3×C42⋊4S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C4≀C2, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C42⋊4S3, C3×C4≀C2, C3×D6⋊C4, C3×C42⋊4S3
►On 24 points - transitive group 24T627
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 17 24)(14 18 21)(15 19 22)(16 20 23)
(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 16 15 14)(17 20 19 18)(21 24 23 22)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 17 24)(14 18 21)(15 19 22)(16 20 23)
(1 20)(2 23)(3 16)(4 18)(5 21)(6 14)(7 19)(8 22)(9 15)(10 17)(11 24)(12 13)
G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,17,24)(14,18,21)(15,19,22)(16,20,23), (13,14,15,16)(17,18,19,20)(21,22,23,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,16,15,14)(17,20,19,18)(21,24,23,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,17,24)(14,18,21)(15,19,22)(16,20,23), (1,20)(2,23)(3,16)(4,18)(5,21)(6,14)(7,19)(8,22)(9,15)(10,17)(11,24)(12,13)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,17,24)(14,18,21)(15,19,22)(16,20,23), (13,14,15,16)(17,18,19,20)(21,22,23,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,16,15,14)(17,20,19,18)(21,24,23,22), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,17,24)(14,18,21)(15,19,22)(16,20,23), (1,20)(2,23)(3,16)(4,18)(5,21)(6,14)(7,19)(8,22)(9,15)(10,17)(11,24)(12,13) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,17,24),(14,18,21),(15,19,22),(16,20,23)], [(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,16,15,14),(17,20,19,18),(21,24,23,22)], [(1,3,2),(4,6,5),(7,9,8),(10,12,11),(13,17,24),(14,18,21),(15,19,22),(16,20,23)], [(1,20),(2,23),(3,16),(4,18),(5,21),(6,14),(7,19),(8,22),(9,15),(10,17),(11,24),(12,13)]])
G:=TransitiveGroup(24,627);
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | ··· | 4G | 4H | 6A | 6B | 6C | ··· | 6M | 6N | 6O | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12AX | 12AY | 12AZ | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 12 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 12 | 12 |
90 irreducible representations
| dim | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D4 | D4 | D6 | C3×S3 | C4×S3 | D12 | C3×D4 | C3⋊D4 | C3×D4 | C4≀C2 | S3×C6 | S3×C12 | C3×D12 | C3×C3⋊D4 | C42⋊4S3 | C3×C4≀C2 | C3×C42⋊4S3 |
| kernel | C3×C42⋊4S3 | C3×C4.Dic3 | C122 | C3×C4○D12 | C42⋊4S3 | C3×Dic6 | C3×D12 | C4.Dic3 | C4×C12 | C4○D12 | Dic6 | D12 | C4×C12 | C3×C12 | C62 | C2×C12 | C42 | C12 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C42⋊4S3 ►in GL2(𝔽13) generated by
| 3 | 0 |
| 0 | 3 |
| 3 | 12 |
| 9 | 3 |
| 0 | 9 |
| 10 | 0 |
| 6 | 8 |
| 6 | 6 |
| 12 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(13))| [3,0,0,3],[3,9,12,3],[0,10,9,0],[6,6,8,6],[12,0,0,1] >;
C3×C42⋊4S3 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_4S_3
% in TeX
G:=Group("C3xC4^2:4S3"); // GroupNames label
G:=SmallGroup(288,239);
// by ID
G=gap.SmallGroup(288,239);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations