direct product, metabelian, supersoluble, monomial
Aliases: C3×C24⋊4S3, (C2×C6)≀C2, C62⋊23D4, C62.211C23, (C23×C6)⋊7S3, C6.63(C6×D4), (C23×C6)⋊11C6, C24⋊10(C3×S3), C32⋊12C22≀C2, C23.35(S3×C6), (C22×C62)⋊3C2, C6.D4⋊13C6, (C22×C6).131D6, (C6×Dic3)⋊19C22, (C2×C62).103C22, (C2×C6)⋊11(C3×D4), (C2×C3⋊D4)⋊8C6, (S3×C2×C6)⋊5C22, C3⋊3(C3×C22≀C2), (C6×C3⋊D4)⋊22C2, C2.26(C6×C3⋊D4), C22⋊5(C3×C3⋊D4), C22.66(S3×C2×C6), (C2×C6)⋊16(C3⋊D4), (C22×S3)⋊2(C2×C6), (C2×Dic3)⋊3(C2×C6), (C3×C6).271(C2×D4), C6.164(C2×C3⋊D4), (C22×C6).67(C2×C6), (C2×C6).66(C22×C6), (C3×C6.D4)⋊29C2, (C2×C6).344(C22×S3), SmallGroup(288,724)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊4S3
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >
Subgroups: 714 in 327 conjugacy classes, 82 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22≀C2, C3×Dic3, S3×C6, C62, C62, C62, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C6×D4, C23×C6, C23×C6, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C2×C62, C24⋊4S3, C3×C22≀C2, C3×C6.D4, C6×C3⋊D4, C22×C62, C3×C24⋊4S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C22≀C2, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C24⋊4S3, C3×C22≀C2, C6×C3⋊D4, C3×C24⋊4S3
►On 24 points - transitive group 24T626
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
G:=sub<Sym(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), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,3,2),(4,6,5),(7,9,8),(10,12,11),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)]])
G:=TransitiveGroup(24,626);
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | ··· | 6F | 6G | ··· | 6BK | 6BL | 6BM | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | ··· | 12 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×C3⋊D4 |
| kernel | C3×C24⋊4S3 | C3×C6.D4 | C6×C3⋊D4 | C22×C62 | C24⋊4S3 | C6.D4 | C2×C3⋊D4 | C23×C6 | C23×C6 | C62 | C22×C6 | C24 | C2×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 1 | 6 | 3 | 2 | 12 | 12 | 6 | 24 |
Matrix representation of C3×C24⋊4S3 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C24⋊4S3 in GAP, Magma, Sage, TeX
C_3\times C_2^4\rtimes_4S_3
% in TeX
G:=Group("C3xC2^4:4S3"); // GroupNames label
G:=SmallGroup(288,724);
// by ID
G=gap.SmallGroup(288,724);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,g*b*g=b*e=e*b,b*f=f*b,g*c*g=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g=f^-1>;
// generators/relations