direct product, non-abelian, soluble, monomial
Aliases: C2xC33:D4, S32:2D6, C6:2S3wrC2, C33:3(C2xD4), (C32xC6):2D4, C33:C4:3C22, C32:4D6:2C22, (C2xS32):5S3, (S32xC6):10C2, C3:3(C2xS3wrC2), (C3xC3:S3):5D4, (C3xS32):3C22, C3:S3:2(C3:D4), (C3xC6):3(C3:D4), (C2xC3:S3).21D6, C32:4(C2xC3:D4), (C2xC33:C4):6C2, C3:S3.3(C22xS3), (C3xC3:S3).6C23, (C2xC32:4D6):5C2, (C6xC3:S3).34C22, SmallGroup(432,755)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC33:D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=c-1, fbf=c, cd=dc, ece-1=fcf=b, ede-1=fdf=d-1, fef=e-1 >
Subgroups: 1396 in 192 conjugacy classes, 31 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, D4, C23, C32, C32, Dic3, D6, C2xC6, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C2xDic3, C3:D4, C22xS3, C22xC6, C33, C32:C4, S32, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, C2xC3:D4, S3xC32, C3xC3:S3, C3xC3:S3, C32xC6, S3wrC2, C2xC32:C4, C2xS32, C2xS32, S3xC2xC6, C33:C4, C3xS32, C3xS32, C32:4D6, C32:4D6, S3xC3xC6, C6xC3:S3, C6xC3:S3, C2xS3wrC2, C33:D4, C2xC33:C4, S32xC6, C2xC32:4D6, C2xC33:D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C2xC3:D4, S3wrC2, C2xS3wrC2, C33:D4, C2xC33:D4
►On 24 points - transitive group 24T1292
(1 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 21)(10 22)(11 23)(12 24)
(2 6 10)(4 12 8)(14 20 24)(16 22 18)
(1 9 5)(3 7 11)(13 23 19)(15 17 21)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 19 23)(14 24 20)(15 17 21)(16 22 18)
(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 20)(2 19)(3 18)(4 17)(5 14)(6 13)(7 16)(8 15)(9 24)(10 23)(11 22)(12 21)
G:=sub<Sym(24)| (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,21)(10,22)(11,23)(12,24), (2,6,10)(4,12,8)(14,20,24)(16,22,18), (1,9,5)(3,7,11)(13,23,19)(15,17,21), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,19,23)(14,24,20)(15,17,21)(16,22,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,20)(2,19)(3,18)(4,17)(5,14)(6,13)(7,16)(8,15)(9,24)(10,23)(11,22)(12,21)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,21)(10,22)(11,23)(12,24), (2,6,10)(4,12,8)(14,20,24)(16,22,18), (1,9,5)(3,7,11)(13,23,19)(15,17,21), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,19,23)(14,24,20)(15,17,21)(16,22,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,20)(2,19)(3,18)(4,17)(5,14)(6,13)(7,16)(8,15)(9,24)(10,23)(11,22)(12,21) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,15),(6,16),(7,13),(8,14),(9,21),(10,22),(11,23),(12,24)], [(2,6,10),(4,12,8),(14,20,24),(16,22,18)], [(1,9,5),(3,7,11),(13,23,19),(15,17,21)], [(1,9,5),(2,6,10),(3,11,7),(4,8,12),(13,19,23),(14,24,20),(15,17,21),(16,22,18)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,20),(2,19),(3,18),(4,17),(5,14),(6,13),(7,16),(8,15),(9,24),(10,23),(11,22),(12,21)]])
G:=TransitiveGroup(24,1292);
►On 24 points - transitive group 24T1312
(1 6)(2 7)(3 8)(4 5)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)
(1 21 20)(2 22 17)(3 18 23)(4 19 24)(5 9 13)(6 14 10)(7 15 11)(8 12 16)
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 9 13)(6 10 14)(7 15 11)(8 16 12)
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 13 9)(6 10 14)(7 15 11)(8 12 16)
(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 6)(2 5)(3 8)(4 7)(9 22)(10 21)(11 24)(12 23)(13 17)(14 20)(15 19)(16 18)
G:=sub<Sym(24)| (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,9,13)(6,14,10)(7,15,11)(8,12,16), (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,9,13)(6,10,14)(7,15,11)(8,16,12), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,13,9)(6,10,14)(7,15,11)(8,12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,5)(3,8)(4,7)(9,22)(10,21)(11,24)(12,23)(13,17)(14,20)(15,19)(16,18)>;
G:=Group( (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,9,13)(6,14,10)(7,15,11)(8,12,16), (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,9,13)(6,10,14)(7,15,11)(8,16,12), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,13,9)(6,10,14)(7,15,11)(8,12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,5)(3,8)(4,7)(9,22)(10,21)(11,24)(12,23)(13,17)(14,20)(15,19)(16,18) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,5),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)], [(1,21,20),(2,22,17),(3,18,23),(4,19,24),(5,9,13),(6,14,10),(7,15,11),(8,12,16)], [(1,20,21),(2,22,17),(3,23,18),(4,19,24),(5,9,13),(6,10,14),(7,15,11),(8,16,12)], [(1,20,21),(2,22,17),(3,18,23),(4,24,19),(5,13,9),(6,10,14),(7,15,11),(8,12,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6),(2,5),(3,8),(4,7),(9,22),(10,21),(11,24),(12,23),(13,17),(14,20),(15,19),(16,18)]])
G:=TransitiveGroup(24,1312);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6P | 6Q | 6R | 6S | 6T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 18 | 18 | 2 | 4 | 4 | 4 | 4 | 8 | 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 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3:D4 | C3:D4 | S3wrC2 | C2xS3wrC2 | C33:D4 | C2xC33:D4 | C33:D4 | C2xC33:D4 |
| kernel | C2xC33:D4 | C33:D4 | C2xC33:C4 | S32xC6 | C2xC32:4D6 | C2xS32 | C3xC3:S3 | C32xC6 | S32 | C2xC3:S3 | C3:S3 | C3xC6 | C6 | C3 | C2 | C1 | C2 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C2xC33:D4 ►in GL4(F7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 4 | 0 |
| 5 | 6 | 1 | 4 |
| 4 | 4 | 0 | 6 |
| 0 | 0 | 0 | 1 |
| 5 | 3 | 5 | 3 |
| 3 | 5 | 2 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 1 | 4 | 5 |
| 1 | 3 | 3 | 5 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 |
| 6 | 2 | 6 | 5 |
| 1 | 2 | 1 | 4 |
| 5 | 2 | 1 | 5 |
| 1 | 1 | 3 | 5 |
| 5 | 3 | 3 | 4 |
| 5 | 5 | 6 | 1 |
| 2 | 5 | 6 | 2 |
| 1 | 1 | 3 | 5 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,1,5,1,2,2,2,1,6,1,1,3,5,4,5,5],[5,5,2,1,3,5,5,1,3,6,6,3,4,1,2,5] >;
C2xC33:D4 in GAP, Magma, Sage, TeX
C_2\times C_3^3\rtimes D_4
% in TeX
G:=Group("C2xC3^3:D4"); // GroupNames label
G:=SmallGroup(432,755);
// by ID
G=gap.SmallGroup(432,755);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,1684,571,165,677,530,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=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,b*c=c*b,b*d=d*b,e*b*e^-1=c^-1,f*b*f=c,c*d=d*c,e*c*e^-1=f*c*f=b,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations