non-abelian, soluble, monomial
Aliases: S32:Dic3, C6.13S3wrC2, (C32xC6).7D4, C33:4(C22:C4), C2.2(C33:D4), C32:2(C6.D4), (C2xS32).S3, (C3xS32):2C4, C3:3(S32:C4), (S32xC6).2C2, C33:9(C2xC4):7C2, (C3xC3:S3).10D4, (C2xC3:S3).11D6, (C2xC33:C4):3C2, C3:S3.3(C3:D4), (C6xC3:S3).7C22, C3:S3.2(C2xDic3), (C3xC6).13(C3:D4), (C3xC3:S3).9(C2xC4), SmallGroup(432,580)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S32:Dic3
G = < a,b,c,d,e,f | a3=b2=c3=d2=e6=1, f2=e3, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=c, bc=cb, bd=db, be=eb, fbf-1=d, dcd=c-1, ce=ec, fcf-1=a, de=ed, fdf-1=b, fef-1=e-1 >
Subgroups: 804 in 132 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, D6, C2xC6, C22:C4, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C22xS3, C22xC6, C33, C3xDic3, C3:Dic3, C32:C4, S32, S32, S3xC6, C2xC3:S3, C62, C6.D4, S3xC32, C3xC3:S3, C32xC6, S3xDic3, C6.D6, C2xC32:C4, C2xS32, S3xC2xC6, C3xC3:Dic3, C33:C4, C3xS32, C3xS32, S3xC3xC6, C6xC3:S3, S32:C4, C33:9(C2xC4), C2xC33:C4, S32xC6, S32:Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C2xDic3, C3:D4, C6.D4, S3wrC2, S32:C4, C33:D4, S32:Dic3
►On 24 points - transitive group 24T1293
(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)
(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 16 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)
G:=sub<Sym(24)| (13,15,17)(14,16,18)(19,23,21)(20,24,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21)>;
G:=Group( (13,15,17)(14,16,18)(19,23,21)(20,24,22), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21) );
G=PermutationGroup([[(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,4,13),(2,15,5,18),(3,14,6,17),(7,20,10,23),(8,19,11,22),(9,24,12,21)]])
G:=TransitiveGroup(24,1293);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6P | 6Q | 6R | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 2 | 4 | 4 | 4 | 4 | 8 | 18 | 18 | 54 | 54 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 12 | ··· | 12 | 18 | 18 | 36 | 36 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | Dic3 | D6 | C3:D4 | C3:D4 | S3wrC2 | S32:C4 | S32:C4 | C33:D4 | S32:Dic3 | C33:D4 | S32:Dic3 |
| kernel | S32:Dic3 | C33:9(C2xC4) | C2xC33:C4 | S32xC6 | C3xS32 | C2xS32 | C3xC3:S3 | C32xC6 | S32 | C2xC3:S3 | C3:S3 | C3xC6 | C6 | C3 | C3 | C2 | C1 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 1 | 1 |
Matrix representation of S32:Dic3 ►in GL4(F7) generated by
| 1 | 0 | 4 | 0 |
| 5 | 6 | 1 | 4 |
| 4 | 4 | 0 | 6 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 4 | 5 |
| 6 | 5 | 3 | 2 |
| 4 | 4 | 1 | 6 |
| 0 | 0 | 0 | 6 |
| 5 | 3 | 5 | 3 |
| 3 | 5 | 2 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 2 | 1 | 5 |
| 4 | 2 | 5 | 4 |
| 0 | 0 | 6 | 0 |
| 1 | 6 | 3 | 0 |
| 4 | 1 | 4 | 5 |
| 1 | 4 | 3 | 5 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 3 |
| 5 | 6 | 5 | 6 |
| 4 | 4 | 1 | 0 |
| 6 | 1 | 4 | 4 |
| 3 | 3 | 2 | 1 |
G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[0,6,4,0,1,5,4,0,4,3,1,0,5,2,6,6],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[4,4,0,1,2,2,0,6,1,5,6,3,5,4,0,0],[4,1,0,0,1,4,0,0,4,3,5,0,5,5,0,3],[5,4,6,3,6,4,1,3,5,1,4,2,6,0,4,1] >;
S32:Dic3 in GAP, Magma, Sage, TeX
S_3^2\rtimes {\rm Dic}_3 % in TeX
G:=Group("S3^2:Dic3"); // GroupNames label
G:=SmallGroup(432,580);
// by ID
G=gap.SmallGroup(432,580);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,1684,571,298,677,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^2=e^6=1,f^2=e^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=c,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=d,d*c*d=c^-1,c*e=e*c,f*c*f^-1=a,d*e=e*d,f*d*f^-1=b,f*e*f^-1=e^-1>;
// generators/relations