direct product, non-abelian, soluble, monomial
Aliases: C3×C32⋊2SD16, C33⋊5SD16, C6.22S3≀C2, D6⋊S3.C6, C32⋊2C8⋊2C6, C32⋊2Q8⋊1C6, (C32×C6).4D4, C32⋊2(C3×SD16), C2.4(C3×S3≀C2), (C3×C6).4(C3×D4), (C3×C32⋊2C8)⋊8C2, C3⋊Dic3.6(C2×C6), (C3×D6⋊S3).2C2, (C3×C32⋊2Q8)⋊12C2, (C3×C3⋊Dic3).32C22, SmallGroup(432,577)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊2SD16
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-1, ebe=b-1, dcd-1=b, ce=ec, ede=d3 >
Subgroups: 396 in 84 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C24, Dic6, C3⋊D4, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C3×SD16, S3×C32, C32×C6, C32⋊2C8, D6⋊S3, C32⋊2Q8, C3×Dic6, C3×C3⋊D4, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C32⋊2SD16, C3×C32⋊2C8, C3×D6⋊S3, C3×C32⋊2Q8, C3×C32⋊2SD16
Quotients: C1, C2, C3, C22, C6, D4, C2×C6, SD16, C3×D4, C3×SD16, S3≀C2, C32⋊2SD16, C3×S3≀C2, C3×C32⋊2SD16
►On 24 points - transitive group 24T1319
(1 21 15)(2 22 16)(3 23 9)(4 24 10)(5 17 11)(6 18 12)(7 19 13)(8 20 14)
(2 22 16)(4 10 24)(6 18 12)(8 14 20)
(1 15 21)(3 23 9)(5 11 17)(7 19 13)
(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)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)
G:=sub<Sym(24)| (1,21,15)(2,22,16)(3,23,9)(4,24,10)(5,17,11)(6,18,12)(7,19,13)(8,20,14), (2,22,16)(4,10,24)(6,18,12)(8,14,20), (1,15,21)(3,23,9)(5,11,17)(7,19,13), (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)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)>;
G:=Group( (1,21,15)(2,22,16)(3,23,9)(4,24,10)(5,17,11)(6,18,12)(7,19,13)(8,20,14), (2,22,16)(4,10,24)(6,18,12)(8,14,20), (1,15,21)(3,23,9)(5,11,17)(7,19,13), (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)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24) );
G=PermutationGroup([[(1,21,15),(2,22,16),(3,23,9),(4,24,10),(5,17,11),(6,18,12),(7,19,13),(8,20,14)], [(2,22,16),(4,10,24),(6,18,12),(8,14,20)], [(1,15,21),(3,23,9),(5,11,17),(7,19,13)], [(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),(3,7),(6,8),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24)]])
G:=TransitiveGroup(24,1319);
45 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 12 | 1 | 1 | 4 | ··· | 4 | 12 | 18 | 1 | 1 | 4 | ··· | 4 | 12 | ··· | 12 | 18 | 18 | 12 | ··· | 12 | 18 | 18 | 18 | 18 | 18 | 18 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | SD16 | C3×D4 | C3×SD16 | S3≀C2 | C32⋊2SD16 | C32⋊2SD16 | C3×S3≀C2 | C3×C32⋊2SD16 |
| kernel | C3×C32⋊2SD16 | C3×C32⋊2C8 | C3×D6⋊S3 | C3×C32⋊2Q8 | C32⋊2SD16 | C32⋊2C8 | D6⋊S3 | C32⋊2Q8 | C32×C6 | C33 | C3×C6 | C32 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of C3×C32⋊2SD16 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 5 | 4 | 2 |
| 5 | 4 | 0 | 5 |
| 5 | 4 | 0 | 3 |
| 3 | 0 | 6 | 2 |
| 4 | 3 | 1 | 4 |
| 0 | 4 | 3 | 4 |
| 6 | 0 | 4 | 0 |
| 4 | 5 | 0 | 3 |
| 1 | 0 | 1 | 6 |
| 6 | 6 | 4 | 2 |
| 3 | 3 | 5 | 0 |
| 0 | 4 | 3 | 2 |
| 5 | 2 | 6 | 2 |
| 1 | 2 | 3 | 2 |
| 3 | 4 | 6 | 2 |
| 6 | 2 | 6 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,5,5,3,5,4,4,0,4,0,0,6,2,5,3,2],[4,0,6,4,3,4,0,5,1,3,4,0,4,4,0,3],[1,6,3,0,0,6,3,4,1,4,5,3,6,2,0,2],[5,1,3,6,2,2,4,2,6,3,6,6,2,2,2,1] >;
C3×C32⋊2SD16 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_2{\rm SD}_{16} % in TeX
G:=Group("C3xC3^2:2SD16"); // GroupNames label
G:=SmallGroup(432,577);
// by ID
G=gap.SmallGroup(432,577);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,197,176,1011,514,80,4037,3036,362,1189,1203]);
// 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^-1,e*b*e=b^-1,d*c*d^-1=b,c*e=e*c,e*d*e=d^3>;
// generators/relations