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 [×2], C3, C3 [×2], C4, C4 [×5], C22, C22 [×2], C6, C6 [×4], C2×C4 [×6], Q8 [×6], C23, C32, Dic3 [×2], Dic3 [×2], C12, C12 [×5], A4, A4, C2×C6, C2×C6 [×2], C22×C4, C22×C4 [×2], C2×Q8 [×4], C3×C6, Dic6, Dic6 [×5], C2×Dic3 [×4], C2×C12 [×2], C3×Q8, C2×A4, C2×A4, C22×C6, C22×Q8, C3×Dic3 [×2], C3×C12, C3×A4, C4×A4, C4×A4 [×3], C2×Dic6 [×4], C22×Dic3 [×2], C22×C12, C3×Dic6, C6×A4, Q8×A4, C22×Dic6, Dic3×A4 [×2], C12×A4, A4×Dic6
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], Q8, A4, D6, C2×C6, C3×S3, Dic6, C3×Q8, C2×A4 [×3], S3×C6, C22×A4, C3×Dic6, S3×A4, Q8×A4, C2×S3×A4, A4×Dic6
►On 72 points
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(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 19 68)(2 20 69)(3 21 70)(4 22 71)(5 23 72)(6 24 61)(7 13 62)(8 14 63)(9 15 64)(10 16 65)(11 17 66)(12 18 67)(25 38 52)(26 39 53)(27 40 54)(28 41 55)(29 42 56)(30 43 57)(31 44 58)(32 45 59)(33 46 60)(34 47 49)(35 48 50)(36 37 51)
(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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)(49 61 55 67)(50 72 56 66)(51 71 57 65)(52 70 58 64)(53 69 59 63)(54 68 60 62)
G:=sub<Sym(72)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,19,68)(2,20,69)(3,21,70)(4,22,71)(5,23,72)(6,24,61)(7,13,62)(8,14,63)(9,15,64)(10,16,65)(11,17,66)(12,18,67)(25,38,52)(26,39,53)(27,40,54)(28,41,55)(29,42,56)(30,43,57)(31,44,58)(32,45,59)(33,46,60)(34,47,49)(35,48,50)(36,37,51), (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), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,40,19,46)(14,39,20,45)(15,38,21,44)(16,37,22,43)(17,48,23,42)(18,47,24,41)(49,61,55,67)(50,72,56,66)(51,71,57,65)(52,70,58,64)(53,69,59,63)(54,68,60,62)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,19,68)(2,20,69)(3,21,70)(4,22,71)(5,23,72)(6,24,61)(7,13,62)(8,14,63)(9,15,64)(10,16,65)(11,17,66)(12,18,67)(25,38,52)(26,39,53)(27,40,54)(28,41,55)(29,42,56)(30,43,57)(31,44,58)(32,45,59)(33,46,60)(34,47,49)(35,48,50)(36,37,51), (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), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,40,19,46)(14,39,20,45)(15,38,21,44)(16,37,22,43)(17,48,23,42)(18,47,24,41)(49,61,55,67)(50,72,56,66)(51,71,57,65)(52,70,58,64)(53,69,59,63)(54,68,60,62) );
G=PermutationGroup([(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72)], [(1,19,68),(2,20,69),(3,21,70),(4,22,71),(5,23,72),(6,24,61),(7,13,62),(8,14,63),(9,15,64),(10,16,65),(11,17,66),(12,18,67),(25,38,52),(26,39,53),(27,40,54),(28,41,55),(29,42,56),(30,43,57),(31,44,58),(32,45,59),(33,46,60),(34,47,49),(35,48,50),(36,37,51)], [(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)], [(1,33,7,27),(2,32,8,26),(3,31,9,25),(4,30,10,36),(5,29,11,35),(6,28,12,34),(13,40,19,46),(14,39,20,45),(15,38,21,44),(16,37,22,43),(17,48,23,42),(18,47,24,41),(49,61,55,67),(50,72,56,66),(51,71,57,65),(52,70,58,64),(53,69,59,63),(54,68,60,62)])
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