direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C3×C12, C62⋊12C12, (C22×C4)⋊C33, C6.18(C6×A4), (C6×A4).12C6, (C22×C12)⋊C32, C22⋊(C32×C12), C23.(C32×C6), (C2×C62).19C6, (C2×C6×C12)⋊3C3, C2.1(A4×C3×C6), (A4×C3×C6).6C2, (C2×C6)⋊2(C3×C12), (C2×A4).2(C3×C6), (C3×C6).25(C2×A4), (C22×C6).12(C3×C6), SmallGroup(432,697)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C3×C12 |
Generators and relations for A4×C3×C12
G = < a,b,c,d,e | a3=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: 492 in 210 conjugacy classes, 102 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, C12, C12, A4, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C2×C12, C2×A4, C22×C6, C33, C3×C12, C3×C12, C3×A4, C62, C62, C4×A4, C22×C12, C32×C6, C6×C12, C6×A4, C2×C62, C32×C12, C32×A4, C12×A4, C2×C6×C12, A4×C3×C6, A4×C3×C12
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3×C6, C2×A4, C33, C3×C12, C3×A4, C4×A4, C32×C6, C6×A4, C32×C12, C32×A4, C12×A4, A4×C3×C6, A4×C3×C12
►On 108 points
Generators in S108
(1 76 40)(2 77 41)(3 78 42)(4 79 43)(5 80 44)(6 81 45)(7 82 46)(8 83 47)(9 84 48)(10 73 37)(11 74 38)(12 75 39)(13 52 95)(14 53 96)(15 54 85)(16 55 86)(17 56 87)(18 57 88)(19 58 89)(20 59 90)(21 60 91)(22 49 92)(23 50 93)(24 51 94)(25 98 71)(26 99 72)(27 100 61)(28 101 62)(29 102 63)(30 103 64)(31 104 65)(32 105 66)(33 106 67)(34 107 68)(35 108 69)(36 97 70)
(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)
(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)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(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)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 69 91)(2 70 92)(3 71 93)(4 72 94)(5 61 95)(6 62 96)(7 63 85)(8 64 86)(9 65 87)(10 66 88)(11 67 89)(12 68 90)(13 80 27)(14 81 28)(15 82 29)(16 83 30)(17 84 31)(18 73 32)(19 74 33)(20 75 34)(21 76 35)(22 77 36)(23 78 25)(24 79 26)(37 105 57)(38 106 58)(39 107 59)(40 108 60)(41 97 49)(42 98 50)(43 99 51)(44 100 52)(45 101 53)(46 102 54)(47 103 55)(48 104 56)
G:=sub<Sym(108)| (1,76,40)(2,77,41)(3,78,42)(4,79,43)(5,80,44)(6,81,45)(7,82,46)(8,83,47)(9,84,48)(10,73,37)(11,74,38)(12,75,39)(13,52,95)(14,53,96)(15,54,85)(16,55,86)(17,56,87)(18,57,88)(19,58,89)(20,59,90)(21,60,91)(22,49,92)(23,50,93)(24,51,94)(25,98,71)(26,99,72)(27,100,61)(28,101,62)(29,102,63)(30,103,64)(31,104,65)(32,105,66)(33,106,67)(34,107,68)(35,108,69)(36,97,70), (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), (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)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(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)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,69,91)(2,70,92)(3,71,93)(4,72,94)(5,61,95)(6,62,96)(7,63,85)(8,64,86)(9,65,87)(10,66,88)(11,67,89)(12,68,90)(13,80,27)(14,81,28)(15,82,29)(16,83,30)(17,84,31)(18,73,32)(19,74,33)(20,75,34)(21,76,35)(22,77,36)(23,78,25)(24,79,26)(37,105,57)(38,106,58)(39,107,59)(40,108,60)(41,97,49)(42,98,50)(43,99,51)(44,100,52)(45,101,53)(46,102,54)(47,103,55)(48,104,56)>;
G:=Group( (1,76,40)(2,77,41)(3,78,42)(4,79,43)(5,80,44)(6,81,45)(7,82,46)(8,83,47)(9,84,48)(10,73,37)(11,74,38)(12,75,39)(13,52,95)(14,53,96)(15,54,85)(16,55,86)(17,56,87)(18,57,88)(19,58,89)(20,59,90)(21,60,91)(22,49,92)(23,50,93)(24,51,94)(25,98,71)(26,99,72)(27,100,61)(28,101,62)(29,102,63)(30,103,64)(31,104,65)(32,105,66)(33,106,67)(34,107,68)(35,108,69)(36,97,70), (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), (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)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(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)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,69,91)(2,70,92)(3,71,93)(4,72,94)(5,61,95)(6,62,96)(7,63,85)(8,64,86)(9,65,87)(10,66,88)(11,67,89)(12,68,90)(13,80,27)(14,81,28)(15,82,29)(16,83,30)(17,84,31)(18,73,32)(19,74,33)(20,75,34)(21,76,35)(22,77,36)(23,78,25)(24,79,26)(37,105,57)(38,106,58)(39,107,59)(40,108,60)(41,97,49)(42,98,50)(43,99,51)(44,100,52)(45,101,53)(46,102,54)(47,103,55)(48,104,56) );
G=PermutationGroup([[(1,76,40),(2,77,41),(3,78,42),(4,79,43),(5,80,44),(6,81,45),(7,82,46),(8,83,47),(9,84,48),(10,73,37),(11,74,38),(12,75,39),(13,52,95),(14,53,96),(15,54,85),(16,55,86),(17,56,87),(18,57,88),(19,58,89),(20,59,90),(21,60,91),(22,49,92),(23,50,93),(24,51,94),(25,98,71),(26,99,72),(27,100,61),(28,101,62),(29,102,63),(30,103,64),(31,104,65),(32,105,66),(33,106,67),(34,107,68),(35,108,69),(36,97,70)], [(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)], [(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),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(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),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,69,91),(2,70,92),(3,71,93),(4,72,94),(5,61,95),(6,62,96),(7,63,85),(8,64,86),(9,65,87),(10,66,88),(11,67,89),(12,68,90),(13,80,27),(14,81,28),(15,82,29),(16,83,30),(17,84,31),(18,73,32),(19,74,33),(20,75,34),(21,76,35),(22,77,36),(23,78,25),(24,79,26),(37,105,57),(38,106,58),(39,107,59),(40,108,60),(41,97,49),(42,98,50),(43,99,51),(44,100,52),(45,101,53),(46,102,54),(47,103,55),(48,104,56)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Z | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 6I | ··· | 6X | 6Y | ··· | 6AP | 12A | ··· | 12P | 12Q | ··· | 12AF | 12AG | ··· | 12BP |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | A4 | C2×A4 | C3×A4 | C4×A4 | C6×A4 | C12×A4 |
| kernel | A4×C3×C12 | A4×C3×C6 | C12×A4 | C2×C6×C12 | C32×A4 | C6×A4 | C2×C62 | C3×A4 | C62 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 24 | 2 | 2 | 24 | 2 | 48 | 4 | 1 | 1 | 8 | 2 | 8 | 16 |
Matrix representation of A4×C3×C12 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 11 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 10 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 3 | 9 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 12 | 10 | 8 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[11,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,10,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,3,0,0,12,9,0,0,0,1],[9,0,0,0,0,0,12,0,0,9,10,0,0,0,8,3] >;
A4×C3×C12 in GAP, Magma, Sage, TeX
A_4\times C_3\times C_{12} % in TeX
G:=Group("A4xC3xC12"); // GroupNames label
G:=SmallGroup(432,697);
// by ID
G=gap.SmallGroup(432,697);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,-2,2,378,4548,7951]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=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