direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C3⋊Dic3, C62⋊7C12, C3⋊(Dic3×A4), C6.15(S3×A4), C32⋊5(C4×A4), (C6×A4).11S3, (C32×A4)⋊7C4, (C3×A4)⋊5Dic3, (C2×C62).10C6, C2.1(A4×C3⋊S3), (A4×C3×C6).5C2, (C3×C6).20(C2×A4), (C2×C6)⋊3(C3×Dic3), C23.2(C3×C3⋊S3), (C2×A4).2(C3⋊S3), C22⋊2(C3×C3⋊Dic3), (C22×C6).17(C3×S3), (C22×C3⋊Dic3)⋊3C3, SmallGroup(432,627)
Series: Derived ►Chief ►Lower central ►Upper central
| C62 — A4×C3⋊Dic3 |
Generators and relations for A4×C3⋊Dic3
G = < a,b,c,d,e,f | a2=b2=c3=d3=e6=1, f2=e3, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >
Subgroups: 668 in 146 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, A4, A4, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×A4, C2×A4, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×A4, C3×A4, C62, C62, C4×A4, C22×Dic3, C32×C6, C2×C3⋊Dic3, C6×A4, C6×A4, C2×C62, C3×C3⋊Dic3, C32×A4, Dic3×A4, C22×C3⋊Dic3, A4×C3×C6, A4×C3⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, A4, C3×S3, C3⋊S3, C2×A4, C3×Dic3, C3⋊Dic3, C4×A4, C3×C3⋊S3, S3×A4, C3×C3⋊Dic3, Dic3×A4, A4×C3⋊S3, A4×C3⋊Dic3
►On 108 points
Generators in S108
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)(67 70)(68 71)(69 72)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(103 106)(104 107)(105 108)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)(97 100)(98 101)(99 102)(103 106)(104 107)(105 108)
(1 62 44)(2 63 45)(3 64 46)(4 65 47)(5 66 48)(6 61 43)(7 102 84)(8 97 79)(9 98 80)(10 99 81)(11 100 82)(12 101 83)(13 96 78)(14 91 73)(15 92 74)(16 93 75)(17 94 76)(18 95 77)(19 58 37)(20 59 38)(21 60 39)(22 55 40)(23 56 41)(24 57 42)(25 53 35)(26 54 36)(27 49 31)(28 50 32)(29 51 33)(30 52 34)(67 103 85)(68 104 86)(69 105 87)(70 106 88)(71 107 89)(72 108 90)
(1 23 29)(2 24 30)(3 19 25)(4 20 26)(5 21 27)(6 22 28)(7 13 107)(8 14 108)(9 15 103)(10 16 104)(11 17 105)(12 18 106)(31 48 39)(32 43 40)(33 44 41)(34 45 42)(35 46 37)(36 47 38)(49 66 60)(50 61 55)(51 62 56)(52 63 57)(53 64 58)(54 65 59)(67 80 74)(68 81 75)(69 82 76)(70 83 77)(71 84 78)(72 79 73)(85 98 92)(86 99 93)(87 100 94)(88 101 95)(89 102 96)(90 97 91)
(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)
(1 72 4 69)(2 71 5 68)(3 70 6 67)(7 49 10 52)(8 54 11 51)(9 53 12 50)(13 60 16 57)(14 59 17 56)(15 58 18 55)(19 77 22 74)(20 76 23 73)(21 75 24 78)(25 83 28 80)(26 82 29 79)(27 81 30 84)(31 99 34 102)(32 98 35 101)(33 97 36 100)(37 95 40 92)(38 94 41 91)(39 93 42 96)(43 85 46 88)(44 90 47 87)(45 89 48 86)(61 103 64 106)(62 108 65 105)(63 107 66 104)
G:=sub<Sym(108)| (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66)(67,70)(68,71)(69,72)(73,76)(74,77)(75,78)(79,82)(80,83)(81,84)(103,106)(104,107)(105,108), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66)(85,88)(86,89)(87,90)(91,94)(92,95)(93,96)(97,100)(98,101)(99,102)(103,106)(104,107)(105,108), (1,62,44)(2,63,45)(3,64,46)(4,65,47)(5,66,48)(6,61,43)(7,102,84)(8,97,79)(9,98,80)(10,99,81)(11,100,82)(12,101,83)(13,96,78)(14,91,73)(15,92,74)(16,93,75)(17,94,76)(18,95,77)(19,58,37)(20,59,38)(21,60,39)(22,55,40)(23,56,41)(24,57,42)(25,53,35)(26,54,36)(27,49,31)(28,50,32)(29,51,33)(30,52,34)(67,103,85)(68,104,86)(69,105,87)(70,106,88)(71,107,89)(72,108,90), (1,23,29)(2,24,30)(3,19,25)(4,20,26)(5,21,27)(6,22,28)(7,13,107)(8,14,108)(9,15,103)(10,16,104)(11,17,105)(12,18,106)(31,48,39)(32,43,40)(33,44,41)(34,45,42)(35,46,37)(36,47,38)(49,66,60)(50,61,55)(51,62,56)(52,63,57)(53,64,58)(54,65,59)(67,80,74)(68,81,75)(69,82,76)(70,83,77)(71,84,78)(72,79,73)(85,98,92)(86,99,93)(87,100,94)(88,101,95)(89,102,96)(90,97,91), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108), (1,72,4,69)(2,71,5,68)(3,70,6,67)(7,49,10,52)(8,54,11,51)(9,53,12,50)(13,60,16,57)(14,59,17,56)(15,58,18,55)(19,77,22,74)(20,76,23,73)(21,75,24,78)(25,83,28,80)(26,82,29,79)(27,81,30,84)(31,99,34,102)(32,98,35,101)(33,97,36,100)(37,95,40,92)(38,94,41,91)(39,93,42,96)(43,85,46,88)(44,90,47,87)(45,89,48,86)(61,103,64,106)(62,108,65,105)(63,107,66,104)>;
G:=Group( (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66)(67,70)(68,71)(69,72)(73,76)(74,77)(75,78)(79,82)(80,83)(81,84)(103,106)(104,107)(105,108), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66)(85,88)(86,89)(87,90)(91,94)(92,95)(93,96)(97,100)(98,101)(99,102)(103,106)(104,107)(105,108), (1,62,44)(2,63,45)(3,64,46)(4,65,47)(5,66,48)(6,61,43)(7,102,84)(8,97,79)(9,98,80)(10,99,81)(11,100,82)(12,101,83)(13,96,78)(14,91,73)(15,92,74)(16,93,75)(17,94,76)(18,95,77)(19,58,37)(20,59,38)(21,60,39)(22,55,40)(23,56,41)(24,57,42)(25,53,35)(26,54,36)(27,49,31)(28,50,32)(29,51,33)(30,52,34)(67,103,85)(68,104,86)(69,105,87)(70,106,88)(71,107,89)(72,108,90), (1,23,29)(2,24,30)(3,19,25)(4,20,26)(5,21,27)(6,22,28)(7,13,107)(8,14,108)(9,15,103)(10,16,104)(11,17,105)(12,18,106)(31,48,39)(32,43,40)(33,44,41)(34,45,42)(35,46,37)(36,47,38)(49,66,60)(50,61,55)(51,62,56)(52,63,57)(53,64,58)(54,65,59)(67,80,74)(68,81,75)(69,82,76)(70,83,77)(71,84,78)(72,79,73)(85,98,92)(86,99,93)(87,100,94)(88,101,95)(89,102,96)(90,97,91), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108), (1,72,4,69)(2,71,5,68)(3,70,6,67)(7,49,10,52)(8,54,11,51)(9,53,12,50)(13,60,16,57)(14,59,17,56)(15,58,18,55)(19,77,22,74)(20,76,23,73)(21,75,24,78)(25,83,28,80)(26,82,29,79)(27,81,30,84)(31,99,34,102)(32,98,35,101)(33,97,36,100)(37,95,40,92)(38,94,41,91)(39,93,42,96)(43,85,46,88)(44,90,47,87)(45,89,48,86)(61,103,64,106)(62,108,65,105)(63,107,66,104) );
G=PermutationGroup([[(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(49,52),(50,53),(51,54),(55,58),(56,59),(57,60),(61,64),(62,65),(63,66),(67,70),(68,71),(69,72),(73,76),(74,77),(75,78),(79,82),(80,83),(81,84),(103,106),(104,107),(105,108)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(49,52),(50,53),(51,54),(55,58),(56,59),(57,60),(61,64),(62,65),(63,66),(85,88),(86,89),(87,90),(91,94),(92,95),(93,96),(97,100),(98,101),(99,102),(103,106),(104,107),(105,108)], [(1,62,44),(2,63,45),(3,64,46),(4,65,47),(5,66,48),(6,61,43),(7,102,84),(8,97,79),(9,98,80),(10,99,81),(11,100,82),(12,101,83),(13,96,78),(14,91,73),(15,92,74),(16,93,75),(17,94,76),(18,95,77),(19,58,37),(20,59,38),(21,60,39),(22,55,40),(23,56,41),(24,57,42),(25,53,35),(26,54,36),(27,49,31),(28,50,32),(29,51,33),(30,52,34),(67,103,85),(68,104,86),(69,105,87),(70,106,88),(71,107,89),(72,108,90)], [(1,23,29),(2,24,30),(3,19,25),(4,20,26),(5,21,27),(6,22,28),(7,13,107),(8,14,108),(9,15,103),(10,16,104),(11,17,105),(12,18,106),(31,48,39),(32,43,40),(33,44,41),(34,45,42),(35,46,37),(36,47,38),(49,66,60),(50,61,55),(51,62,56),(52,63,57),(53,64,58),(54,65,59),(67,80,74),(68,81,75),(69,82,76),(70,83,77),(71,84,78),(72,79,73),(85,98,92),(86,99,93),(87,100,94),(88,101,95),(89,102,96),(90,97,91)], [(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108)], [(1,72,4,69),(2,71,5,68),(3,70,6,67),(7,49,10,52),(8,54,11,51),(9,53,12,50),(13,60,16,57),(14,59,17,56),(15,58,18,55),(19,77,22,74),(20,76,23,73),(21,75,24,78),(25,83,28,80),(26,82,29,79),(27,81,30,84),(31,99,34,102),(32,98,35,101),(33,97,36,100),(37,95,40,92),(38,94,41,91),(39,93,42,96),(43,85,46,88),(44,90,47,87),(45,89,48,86),(61,103,64,106),(62,108,65,105),(63,107,66,104)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 3G | ··· | 3N | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6N | 6O | ··· | 6V | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 9 | 9 | 27 | 27 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 36 | 36 | 36 | 36 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 | A4 | C2×A4 | C4×A4 | S3×A4 | Dic3×A4 |
| kernel | A4×C3⋊Dic3 | A4×C3×C6 | C22×C3⋊Dic3 | C32×A4 | C2×C62 | C62 | C6×A4 | C3×A4 | C22×C6 | C2×C6 | C3⋊Dic3 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 4 | 4 |
Matrix representation of A4×C3⋊Dic3 ►in GL7(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 3 | 4 | 10 |
| 9 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 10 | 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,12,0,0,0,0,0,11,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,4,0,0,0,0,0,1,10],[9,0,0,0,0,0,0,11,3,0,0,0,0,0,0,0,9,11,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,11,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[10,4,0,0,0,0,0,4,3,0,0,0,0,0,0,0,9,9,0,0,0,0,0,1,4,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5] >;
A4×C3⋊Dic3 in GAP, Magma, Sage, TeX
A_4\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("A4xC3:Dic3"); // GroupNames label
G:=SmallGroup(432,627);
// by ID
G=gap.SmallGroup(432,627);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,2,-3,-3,42,514,221,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^6=1,f^2=e^3,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations