direct product, non-abelian, soluble, monomial
Aliases: C3×C32⋊D8, C33⋊1D8, C32⋊(C3×D8), C6.21S3≀C2, D6⋊S3⋊1C6, C32⋊2C8⋊1C6, (C32×C6).3D4, C2.3(C3×S3≀C2), (C3×C6).3(C3×D4), (C3×D6⋊S3)⋊8C2, (C3×C32⋊2C8)⋊3C2, C3⋊Dic3.5(C2×C6), (C3×C3⋊Dic3).31C22, SmallGroup(432,576)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >
Subgroups: 492 in 96 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, Dic3, C12, D6, C2×C6, D8, C3×S3, C3×C6, C3×C6, C24, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, C62, C3×D8, S3×C32, C32×C6, C32⋊2C8, D6⋊S3, C3×C3⋊D4, C3×C3⋊Dic3, S3×C3×C6, C32⋊D8, C3×C32⋊2C8, C3×D6⋊S3, C3×C32⋊D8
Quotients: C1, C2, C3, C22, C6, D4, C2×C6, D8, C3×D4, C3×D8, S3≀C2, C32⋊D8, C3×S3≀C2, C3×C32⋊D8
►On 24 points - transitive group 24T1318
(1 10 21)(2 11 22)(3 12 23)(4 13 24)(5 14 17)(6 15 18)(7 16 19)(8 9 20)
(1 10 21)(2 11 22)(3 23 12)(4 24 13)(5 14 17)(6 15 18)(7 19 16)(8 20 9)
(1 10 21)(2 22 11)(3 23 12)(4 13 24)(5 14 17)(6 18 15)(7 19 16)(8 9 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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)
G:=sub<Sym(24)| (1,10,21)(2,11,22)(3,12,23)(4,13,24)(5,14,17)(6,15,18)(7,16,19)(8,9,20), (1,10,21)(2,11,22)(3,23,12)(4,24,13)(5,14,17)(6,15,18)(7,19,16)(8,20,9), (1,10,21)(2,22,11)(3,23,12)(4,13,24)(5,14,17)(6,18,15)(7,19,16)(8,9,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,24)(18,23)(19,22)(20,21)>;
G:=Group( (1,10,21)(2,11,22)(3,12,23)(4,13,24)(5,14,17)(6,15,18)(7,16,19)(8,9,20), (1,10,21)(2,11,22)(3,23,12)(4,24,13)(5,14,17)(6,15,18)(7,19,16)(8,20,9), (1,10,21)(2,22,11)(3,23,12)(4,13,24)(5,14,17)(6,18,15)(7,19,16)(8,9,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,24)(18,23)(19,22)(20,21) );
G=PermutationGroup([[(1,10,21),(2,11,22),(3,12,23),(4,13,24),(5,14,17),(6,15,18),(7,16,19),(8,9,20)], [(1,10,21),(2,11,22),(3,23,12),(4,24,13),(5,14,17),(6,15,18),(7,19,16),(8,20,9)], [(1,10,21),(2,22,11),(3,23,12),(4,13,24),(5,14,17),(6,18,15),(7,19,16),(8,9,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,24),(18,23),(19,22),(20,21)]])
G:=TransitiveGroup(24,1318);
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4 | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6X | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 12 | 12 | 1 | 1 | 4 | ··· | 4 | 18 | 1 | 1 | 4 | ··· | 4 | 12 | ··· | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | C3×D8 | S3≀C2 | C32⋊D8 | C3×S3≀C2 | C3×C32⋊D8 |
| kernel | C3×C32⋊D8 | C3×C32⋊2C8 | C3×D6⋊S3 | C32⋊D8 | C32⋊2C8 | D6⋊S3 | C32×C6 | C33 | C3×C6 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C32⋊D8 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 5 | 3 | 5 | 3 |
| 3 | 5 | 2 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 2 | 4 | 3 |
| 4 | 5 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 1 |
| 4 | 2 | 1 | 4 |
| 6 | 6 | 0 | 3 |
| 2 | 5 | 6 | 4 |
| 1 | 1 | 6 | 5 |
| 0 | 2 | 4 | 1 |
| 2 | 4 | 6 | 5 |
| 3 | 3 | 3 | 1 |
| 6 | 1 | 4 | 0 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[4,6,2,1,2,6,5,1,1,0,6,6,4,3,4,5],[0,2,3,6,2,4,3,1,4,6,3,4,1,5,1,0] >;
C3×C32⋊D8 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes D_8
% in TeX
G:=Group("C3xC3^2:D8"); // GroupNames label
G:=SmallGroup(432,576);
// by ID
G=gap.SmallGroup(432,576);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,197,1011,514,80,4037,3036,362,1189,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations