direct product, metabelian, supersoluble, monomial
Aliases: C3xS3xC3:D4, C62:29D6, D6:6(S3xC6), (S3xC6):17D6, C62:9(C2xC6), C32:9(C6xD4), C33:19(C2xD4), C3:D12:5C6, D6:S3:5C6, (S3xC62):3C2, (S3xDic3):3C6, Dic3:1(S3xC6), (S3xC32):6D4, C32:26(S3xD4), C32:7D4:6C6, (C3xDic3):10D6, (C3xC62):1C22, (C32xC6).36C23, (C32xDic3):5C22, (C2xC6):8S32, C3:5(C3xS3xD4), (C2xS32):3C6, (S32xC6):7C2, (S3xC2xC6):5C6, (S3xC2xC6):5S3, C2.17(S32xC6), C22:3(C3xS32), C3:2(C6xC3:D4), C6.17(S3xC2xC6), (S3xC6):6(C2xC6), C6.120(C2xS32), (C2xC6):10(S3xC6), (C3xS3):2(C3xD4), (C3xC3:D4):3C6, (S3xC3xC6):9C22, (C3xS3xDic3):8C2, (C6xC3:S3):7C22, C3:Dic3:3(C2xC6), (C22xS3):5(C3xS3), (C3xDic3):1(C2xC6), (C3xC32:7D4):8C2, (C32xC3:D4):3C2, C32:18(C2xC3:D4), (C3xC3:D12):11C2, (C3xD6:S3):12C2, (C3xC6).27(C22xC6), (C3xC6).141(C22xS3), (C3xC3:Dic3):10C22, (C2xC3:S3):5(C2xC6), SmallGroup(432,658)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xS3xC3:D4
G = < a,b,c,d,e,f | a3=b3=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >
Subgroups: 1032 in 290 conjugacy classes, 72 normal (64 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2xC4, D4, C23, C32, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C62, C62, S3xD4, C2xC3:D4, C6xD4, S3xC32, S3xC32, C3xC3:S3, C32xC6, C32xC6, S3xDic3, D6:S3, C3:D12, S3xC12, C3xD12, C6xDic3, C3xC3:D4, C3xC3:D4, C32:7D4, D4xC32, C2xS32, S3xC2xC6, S3xC2xC6, C2xC62, C32xDic3, C3xC3:Dic3, C3xS32, S3xC3xC6, S3xC3xC6, C6xC3:S3, C3xC62, S3xC3:D4, C3xS3xD4, C6xC3:D4, C3xS3xDic3, C3xD6:S3, C3xC3:D12, C32xC3:D4, C3xC32:7D4, S32xC6, S3xC62, C3xS3xC3:D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3:D4, C3xD4, C22xS3, C22xC6, S32, S3xC6, S3xD4, C2xC3:D4, C6xD4, C3xC3:D4, C2xS32, S3xC2xC6, C3xS32, S3xC3:D4, C3xS3xD4, C6xC3:D4, S32xC6, C3xS3xC3:D4
►On 24 points - transitive group 24T1280
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 17 16)(2 13 18)(3 19 14)(4 15 20)(5 10 23)(6 24 11)(7 12 21)(8 22 9)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
G:=sub<Sym(24)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,10,23)(6,24,11)(7,12,21)(8,22,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)>;
G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,10,23)(6,24,11)(7,12,21)(8,22,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24) );
G=PermutationGroup([[(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(1,17,16),(2,13,18),(3,19,14),(4,15,20),(5,10,23),(6,24,11),(7,12,21),(8,22,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24)]])
G:=TransitiveGroup(24,1280);
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 6A | 6B | 6C | ··· | 6P | 6Q | 6R | 6S | 6T | 6U | ··· | 6AF | 6AG | ··· | 6AV | 6AW | 6AX | 6AY | 6AZ | 6BA | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 18 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 18 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 18 | 18 | 6 | 6 | 12 | 12 | 12 | 18 | 18 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | S3 | D4 | D6 | D6 | D6 | C3xS3 | C3xS3 | C3:D4 | C3xD4 | S3xC6 | S3xC6 | S3xC6 | C3xC3:D4 | S32 | S3xD4 | C2xS32 | C3xS32 | S3xC3:D4 | C3xS3xD4 | S32xC6 | C3xS3xC3:D4 |
| kernel | C3xS3xC3:D4 | C3xS3xDic3 | C3xD6:S3 | C3xC3:D12 | C32xC3:D4 | C3xC32:7D4 | S32xC6 | S3xC62 | S3xC3:D4 | S3xDic3 | D6:S3 | C3:D12 | C3xC3:D4 | C32:7D4 | C2xS32 | S3xC2xC6 | C3xC3:D4 | S3xC2xC6 | S3xC32 | C3xDic3 | S3xC6 | C62 | C3:D4 | C22xS3 | C3xS3 | C3xS3 | Dic3 | D6 | C2xC6 | S3 | C2xC6 | C32 | C6 | C22 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 4 | 4 | 2 | 6 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of C3xS3xC3:D4 ►in GL4(F7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 6 | 0 | 3 | 1 |
| 5 | 3 | 6 | 0 |
| 1 | 6 | 4 | 5 |
| 3 | 3 | 2 | 6 |
| 3 | 3 | 1 | 5 |
| 2 | 5 | 2 | 0 |
| 6 | 6 | 1 | 2 |
| 3 | 1 | 5 | 5 |
| 4 | 1 | 2 | 1 |
| 3 | 3 | 6 | 4 |
| 6 | 6 | 4 | 2 |
| 6 | 1 | 4 | 1 |
| 3 | 1 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 4 | 5 | 6 | 3 |
| 0 | 3 | 4 | 1 |
| 6 | 3 | 5 | 0 |
| 2 | 4 | 4 | 6 |
| 3 | 1 | 0 | 2 |
| 4 | 4 | 5 | 4 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[3,2,6,3,3,5,6,1,1,2,1,5,5,0,2,5],[4,3,6,6,1,3,6,1,2,6,4,4,1,4,2,1],[3,4,4,0,1,4,5,3,0,0,6,4,0,0,3,1],[6,2,3,4,3,4,1,4,5,4,0,5,0,6,2,4] >;
C3xS3xC3:D4 in GAP, Magma, Sage, TeX
C_3\times S_3\times C_3\rtimes D_4
% in TeX
G:=Group("C3xS3xC3:D4"); // GroupNames label
G:=SmallGroup(432,658);
// by ID
G=gap.SmallGroup(432,658);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations