direct product, metabelian, soluble, monomial
Aliases: A4×Dic6, C3⋊(Q8×A4), C4.1(S3×A4), (C3×A4)⋊4Q8, (C4×A4).3S3, C12.1(C2×A4), (C12×A4).3C2, (C2×A4).13D6, C6.1(C22×A4), (C22×Dic6)⋊C3, C23.16(S3×C6), (C22×C12).1C6, Dic3.1(C2×A4), (Dic3×A4).2C2, C22⋊2(C3×Dic6), (C22×Dic3).C6, (C6×A4).18C22, (C2×C6)⋊(C3×Q8), C2.3(C2×S3×A4), (C22×C4).3(C3×S3), (C22×C6).1(C2×C6), SmallGroup(288,918)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for A4×Dic6
G = < a,b,c,d,e | a2=b2=c3=d12=1, e2=d6, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 378 in 98 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, C12, A4, A4, C2×C6, C2×C6, C22×C4, C22×C4, C2×Q8, C3×C6, Dic6, Dic6, C2×Dic3, C2×C12, C3×Q8, C2×A4, C2×A4, C22×C6, C22×Q8, C3×Dic3, C3×C12, C3×A4, C4×A4, C4×A4, C2×Dic6, C22×Dic3, C22×C12, C3×Dic6, C6×A4, Q8×A4, C22×Dic6, Dic3×A4, C12×A4, A4×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, A4, D6, C2×C6, C3×S3, Dic6, C3×Q8, C2×A4, S3×C6, C22×A4, C3×Dic6, S3×A4, Q8×A4, C2×S3×A4, A4×Dic6
►On 72 points
Generators in S72
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(1 63 19)(2 64 20)(3 65 21)(4 66 22)(5 67 23)(6 68 24)(7 69 13)(8 70 14)(9 71 15)(10 72 16)(11 61 17)(12 62 18)(25 55 39)(26 56 40)(27 57 41)(28 58 42)(29 59 43)(30 60 44)(31 49 45)(32 50 46)(33 51 47)(34 52 48)(35 53 37)(36 54 38)
(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)
(1 39 7 45)(2 38 8 44)(3 37 9 43)(4 48 10 42)(5 47 11 41)(6 46 12 40)(13 49 19 55)(14 60 20 54)(15 59 21 53)(16 58 22 52)(17 57 23 51)(18 56 24 50)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)
G:=sub<Sym(72)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,63,19)(2,64,20)(3,65,21)(4,66,22)(5,67,23)(6,68,24)(7,69,13)(8,70,14)(9,71,15)(10,72,16)(11,61,17)(12,62,18)(25,55,39)(26,56,40)(27,57,41)(28,58,42)(29,59,43)(30,60,44)(31,49,45)(32,50,46)(33,51,47)(34,52,48)(35,53,37)(36,54,38), (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), (1,39,7,45)(2,38,8,44)(3,37,9,43)(4,48,10,42)(5,47,11,41)(6,46,12,40)(13,49,19,55)(14,60,20,54)(15,59,21,53)(16,58,22,52)(17,57,23,51)(18,56,24,50)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,63,19)(2,64,20)(3,65,21)(4,66,22)(5,67,23)(6,68,24)(7,69,13)(8,70,14)(9,71,15)(10,72,16)(11,61,17)(12,62,18)(25,55,39)(26,56,40)(27,57,41)(28,58,42)(29,59,43)(30,60,44)(31,49,45)(32,50,46)(33,51,47)(34,52,48)(35,53,37)(36,54,38), (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), (1,39,7,45)(2,38,8,44)(3,37,9,43)(4,48,10,42)(5,47,11,41)(6,46,12,40)(13,49,19,55)(14,60,20,54)(15,59,21,53)(16,58,22,52)(17,57,23,51)(18,56,24,50)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72)], [(1,63,19),(2,64,20),(3,65,21),(4,66,22),(5,67,23),(6,68,24),(7,69,13),(8,70,14),(9,71,15),(10,72,16),(11,61,17),(12,62,18),(25,55,39),(26,56,40),(27,57,41),(28,58,42),(29,59,43),(30,60,44),(31,49,45),(32,50,46),(33,51,47),(34,52,48),(35,53,37),(36,54,38)], [(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)], [(1,39,7,45),(2,38,8,44),(3,37,9,43),(4,48,10,42),(5,47,11,41),(6,46,12,40),(13,49,19,55),(14,60,20,54),(15,59,21,53),(16,58,22,52),(17,57,23,51),(18,56,24,50),(25,69,31,63),(26,68,32,62),(27,67,33,61),(28,66,34,72),(29,65,35,71),(30,64,36,70)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 2 | 4 | 4 | 8 | 8 | 2 | 6 | 6 | 6 | 18 | 18 | 2 | 4 | 4 | 6 | 6 | 8 | 8 | 2 | 2 | 6 | 6 | 8 | ··· | 8 | 24 | 24 | 24 | 24 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | Q8 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | C3×Dic6 | A4 | C2×A4 | C2×A4 | S3×A4 | Q8×A4 | C2×S3×A4 | A4×Dic6 |
| kernel | A4×Dic6 | Dic3×A4 | C12×A4 | C22×Dic6 | C22×Dic3 | C22×C12 | C4×A4 | C3×A4 | C2×A4 | C22×C4 | A4 | C2×C6 | C23 | C22 | Dic6 | Dic3 | C12 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of A4×Dic6 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 | 12 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 10 | 10 | 10 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,10,0,0,0,0,10,0,0,0,3,10],[6,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12] >;
A4×Dic6 in GAP, Magma, Sage, TeX
A_4\times {\rm Dic}_6 % in TeX
G:=Group("A4xDic6"); // GroupNames label
G:=SmallGroup(288,918);
// by ID
G=gap.SmallGroup(288,918);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,84,197,92,648,271,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=1,e^2=d^6,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations