direct product, metabelian, soluble, monomial, A-group
Aliases: C3×Dic3×A4, C62⋊6C12, C3⋊(C12×A4), C6.3(C6×A4), (C3×A4)⋊5C12, (C6×A4).9C6, C6.22(S3×A4), C32⋊4(C4×A4), (C6×A4).12S3, (C32×A4)⋊6C4, (C2×C62).9C6, C23.2(S3×C32), (C22×Dic3)⋊C32, C22⋊2(C32×Dic3), (C2×C6)⋊(C3×C12), C2.1(C3×S3×A4), (Dic3×C2×C6)⋊C3, (A4×C3×C6).4C2, (C22×C6).(C3×C6), (C2×A4).2(C3×S3), (C3×C6).17(C2×A4), (C2×C6)⋊4(C3×Dic3), (C22×C6).20(C3×S3), SmallGroup(432,624)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C6 — C3×Dic3×A4 |
Generators and relations for C3×Dic3×A4
G = < a,b,c,d,e,f | a3=b6=d2=e2=f3=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Subgroups: 484 in 136 conjugacy classes, 40 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, Dic3, C12, A4, A4, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×C6, C22×C6, C33, C3×Dic3, C3×Dic3, C3×C12, C3×A4, C3×A4, C3×A4, C62, C62, C4×A4, C22×Dic3, C22×C12, C32×C6, C6×Dic3, C6×A4, C6×A4, C6×A4, C2×C62, C32×Dic3, C32×A4, Dic3×A4, C12×A4, Dic3×C2×C6, A4×C3×C6, C3×Dic3×A4
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, A4, C3×S3, C3×C6, C2×A4, C3×Dic3, C3×C12, C3×A4, C4×A4, S3×C32, S3×A4, C6×A4, C32×Dic3, Dic3×A4, C12×A4, C3×S3×A4, C3×Dic3×A4
►On 36 points
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(1 15 7)(2 16 8)(3 17 9)(4 18 10)(5 13 11)(6 14 12)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)
G:=sub<Sym(36)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28)>;
G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,15,7)(2,16,8)(3,17,9)(4,18,10)(5,13,11)(6,14,12)(19,33,29)(20,34,30)(21,35,25)(22,36,26)(23,31,27)(24,32,28) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(1,15,7),(2,16,8),(3,17,9),(4,18,10),(5,13,11),(6,14,12),(19,33,29),(20,34,30),(21,35,25),(22,36,26),(23,31,27),(24,32,28)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3K | 3L | ··· | 3Q | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | ··· | 6O | 6P | ··· | 6U | 6V | ··· | 6AA | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 3 | 3 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 12 | ··· | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | C3×S3 | C3×S3 | C3×Dic3 | C3×Dic3 | A4 | C2×A4 | C3×A4 | C4×A4 | C6×A4 | C12×A4 | S3×A4 | Dic3×A4 | C3×S3×A4 | C3×Dic3×A4 |
| kernel | C3×Dic3×A4 | A4×C3×C6 | Dic3×A4 | Dic3×C2×C6 | C32×A4 | C6×A4 | C2×C62 | C3×A4 | C62 | C6×A4 | C3×A4 | C2×A4 | C22×C6 | A4 | C2×C6 | C3×Dic3 | C3×C6 | Dic3 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 4 | 1 | 1 | 6 | 2 | 6 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Dic3×A4 ►in GL5(𝔽13)
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 10 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 3 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[10,3,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,3,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,9,0,0] >;
C3×Dic3×A4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3\times A_4 % in TeX
G:=Group("C3xDic3xA4"); // GroupNames label
G:=SmallGroup(432,624);
// by ID
G=gap.SmallGroup(432,624);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-2,2,-3,126,1901,768,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^6=d^2=e^2=f^3=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=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,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations