direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C2×C6, C22⋊C62, C24⋊2C32, C23⋊(C3×C6), (C22×C6)⋊2C6, (C23×C6)⋊1C3, (C2×C6)⋊3(C2×C6), SmallGroup(144,193)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C2×C6 |
Generators and relations for A4×C2×C6
G = < a,b,c,d,e | a2=b6=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Subgroups: 234 in 98 conjugacy classes, 40 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3, C3 [×3], C22 [×2], C22 [×11], C6 [×3], C6 [×13], C23 [×3], C23 [×4], C32, A4 [×3], C2×C6 [×2], C2×C6 [×14], C24, C3×C6 [×3], C2×A4 [×9], C22×C6 [×3], C22×C6 [×4], C3×A4, C62, C22×A4 [×3], C23×C6, C6×A4 [×3], A4×C2×C6
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, A4, C2×C6 [×4], C3×C6 [×3], C2×A4 [×3], C3×A4, C62, C22×A4, C6×A4 [×3], A4×C2×C6
►On 36 points
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)
(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 4)(2 5)(3 6)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 16)(14 17)(15 18)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 10)(8 11)(9 12)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 34)(32 35)(33 36)
(1 34 24)(2 35 19)(3 36 20)(4 31 21)(5 32 22)(6 33 23)(7 28 18)(8 29 13)(9 30 14)(10 25 15)(11 26 16)(12 27 17)
G:=sub<Sym(36)| (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28), (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), (1,4)(2,5)(3,6)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(13,16)(14,17)(15,18)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,34)(32,35)(33,36), (1,34,24)(2,35,19)(3,36,20)(4,31,21)(5,32,22)(6,33,23)(7,28,18)(8,29,13)(9,30,14)(10,25,15)(11,26,16)(12,27,17)>;
G:=Group( (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28), (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), (1,4)(2,5)(3,6)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(13,16)(14,17)(15,18)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(31,34)(32,35)(33,36), (1,34,24)(2,35,19)(3,36,20)(4,31,21)(5,32,22)(6,33,23)(7,28,18)(8,29,13)(9,30,14)(10,25,15)(11,26,16)(12,27,17) );
G=PermutationGroup([(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(19,29),(20,30),(21,25),(22,26),(23,27),(24,28)], [(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)], [(1,4),(2,5),(3,6),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(13,16),(14,17),(15,18),(19,26),(20,27),(21,28),(22,29),(23,30),(24,25)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,10),(8,11),(9,12),(19,29),(20,30),(21,25),(22,26),(23,27),(24,28),(31,34),(32,35),(33,36)], [(1,34,24),(2,35,19),(3,36,20),(4,31,21),(5,32,22),(6,33,23),(7,28,18),(8,29,13),(9,30,14),(10,25,15),(11,26,16),(12,27,17)])
A4×C2×C6 is a maximal subgroup of
(C2×C6)⋊4S4 C24⋊He3 C24⋊3- 1+2
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3H | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | A4 | C2×A4 | C3×A4 | C6×A4 |
| kernel | A4×C2×C6 | C6×A4 | C22×A4 | C23×C6 | C2×A4 | C22×C6 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 6 | 2 | 18 | 6 | 1 | 3 | 2 | 6 |
Matrix representation of A4×C2×C6 ►in GL4(𝔽7) generated by
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 5 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 3 | 1 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 4 | 6 | 0 |
| 0 | 2 | 0 | 6 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 5 | 0 |
| 0 | 0 | 3 | 1 |
| 0 | 0 | 5 | 0 |
G:=sub<GL(4,GF(7))| [1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[5,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,3,0,0,0,1,0,0,0,0,6],[1,0,0,0,0,1,4,2,0,0,6,0,0,0,0,6],[4,0,0,0,0,4,0,0,0,5,3,5,0,0,1,0] >;
A4×C2×C6 in GAP, Magma, Sage, TeX
A_4\times C_2\times C_6
% in TeX
G:=Group("A4xC2xC6"); // GroupNames label
G:=SmallGroup(144,193);
// by ID
G=gap.SmallGroup(144,193);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,2,556,989]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations