direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C2×C12, C23.6C62, (C23×C12)⋊C3, C22⋊(C6×C12), C24.(C3×C6), (C23×C4)⋊C32, (C22×C12)⋊4C6, (C22×C6)⋊4C12, C23⋊2(C3×C12), C22.7(C6×A4), (C23×C6).7C6, (C22×A4).2C6, C6.20(C22×A4), (C6×A4).23C22, C2.1(A4×C2×C6), (A4×C2×C6).4C2, (C2×C6)⋊6(C2×C12), (C2×A4).6(C2×C6), (C2×C6).28(C2×A4), (C22×C4)⋊2(C3×C6), (C22×C6).39(C2×C6), SmallGroup(288,979)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C2×C12 |
Generators and relations for A4×C2×C12
G = < a,b,c,d,e | a2=b12=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: 396 in 164 conjugacy classes, 64 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C32, C12, C12, A4, C2×C6, C2×C6, C22×C4, C22×C4, C24, C3×C6, C2×C12, C2×C12, C2×A4, C22×C6, C22×C6, C22×C6, C23×C4, C3×C12, C3×A4, C62, C4×A4, C22×C12, C22×C12, C22×A4, C23×C6, C6×C12, C6×A4, C6×A4, C2×C4×A4, C23×C12, C12×A4, A4×C2×C6, A4×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, A4, C2×C6, C3×C6, C2×C12, C2×A4, C3×C12, C3×A4, C62, C4×A4, C22×A4, C6×C12, C6×A4, C2×C4×A4, C12×A4, A4×C2×C6, A4×C2×C12
►On 72 points
Generators in S72
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 72)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)
(1 72 17)(2 61 18)(3 62 19)(4 63 20)(5 64 21)(6 65 22)(7 66 23)(8 67 24)(9 68 13)(10 69 14)(11 70 15)(12 71 16)(25 49 47)(26 50 48)(27 51 37)(28 52 38)(29 53 39)(30 54 40)(31 55 41)(32 56 42)(33 57 43)(34 58 44)(35 59 45)(36 60 46)
G:=sub<Sym(72)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65), (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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56), (1,72,17)(2,61,18)(3,62,19)(4,63,20)(5,64,21)(6,65,22)(7,66,23)(8,67,24)(9,68,13)(10,69,14)(11,70,15)(12,71,16)(25,49,47)(26,50,48)(27,51,37)(28,52,38)(29,53,39)(30,54,40)(31,55,41)(32,56,42)(33,57,43)(34,58,44)(35,59,45)(36,60,46)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65), (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)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56), (1,72,17)(2,61,18)(3,62,19)(4,63,20)(5,64,21)(6,65,22)(7,66,23)(8,67,24)(9,68,13)(10,69,14)(11,70,15)(12,71,16)(25,49,47)(26,50,48)(27,51,37)(28,52,38)(29,53,39)(30,54,40)(31,55,41)(32,56,42)(33,57,43)(34,58,44)(35,59,45)(36,60,46) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,61),(33,62),(34,63),(35,64),(36,65)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,72),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71)], [(1,47),(2,48),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56)], [(1,72,17),(2,61,18),(3,62,19),(4,63,20),(5,64,21),(6,65,22),(7,66,23),(8,67,24),(9,68,13),(10,69,14),(11,70,15),(12,71,16),(25,49,47),(26,50,48),(27,51,37),(28,52,38),(29,53,39),(30,54,40),(31,55,41),(32,56,42),(33,57,43),(34,58,44),(35,59,45),(36,60,46)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6AF | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | A4 | C2×A4 | C2×A4 | C3×A4 | C4×A4 | C6×A4 | C6×A4 | C12×A4 |
| kernel | A4×C2×C12 | C12×A4 | A4×C2×C6 | C2×C4×A4 | C23×C12 | C6×A4 | C4×A4 | C22×C12 | C22×A4 | C23×C6 | C2×A4 | C22×C6 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 6 | 2 | 4 | 12 | 4 | 6 | 2 | 24 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of A4×C2×C12 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 4 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 9 | 3 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 12 | 4 | 7 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,4,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,9,0,0,12,3,0,0,0,1],[1,0,0,0,0,0,12,0,0,3,4,0,0,0,7,9] >;
A4×C2×C12 in GAP, Magma, Sage, TeX
A_4\times C_2\times C_{12} % in TeX
G:=Group("A4xC2xC12"); // GroupNames label
G:=SmallGroup(288,979);
// by ID
G=gap.SmallGroup(288,979);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,260,1531,2666]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=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