direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C2×C12, C62.190C23, C6⋊(C4×C12), C12⋊8(C2×C12), (C6×C12)⋊16C4, (C2×C12)⋊7C12, (C3×C6)⋊3C42, C32⋊7(C2×C42), (C2×C12).463D6, C23.38(S3×C6), C62.79(C2×C4), C6.22(C22×C12), (C22×C12).26C6, (C22×C12).45S3, C22.15(S3×C12), (C22×C6).170D6, (C6×C12).351C22, (C2×C62).93C22, (C22×Dic3).8C6, C22.13(C6×Dic3), C6.42(C22×Dic3), (C6×Dic3).167C22, C3⋊2(C2×C4×C12), C2.3(S3×C2×C12), (C2×C6×C12).21C2, C6.116(S3×C2×C4), (C3×C12)⋊24(C2×C4), C2.2(Dic3×C2×C6), (C2×C6).85(C4×S3), C22.19(S3×C2×C6), (C2×C4).101(S3×C6), (C2×C6).20(C2×C12), (Dic3×C2×C6).14C2, (C2×C12).131(C2×C6), (C2×C6).45(C22×C6), (C22×C4).14(C3×S3), (C22×C6).57(C2×C6), (C3×C6).88(C22×C4), (C2×C6).62(C2×Dic3), (C2×C6).323(C22×S3), (C2×Dic3).49(C2×C6), SmallGroup(288,693)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C2×C12 |
Generators and relations for Dic3×C2×C12
G = < a,b,c,d | a2=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 378 in 243 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22×C4, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C2×C42, C3×Dic3, C3×C12, C62, C62, C4×Dic3, C4×C12, C22×Dic3, C22×C12, C22×C12, C6×Dic3, C6×C12, C2×C62, C2×C4×Dic3, C2×C4×C12, Dic3×C12, Dic3×C2×C6, C2×C6×C12, Dic3×C2×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C42, C22×C4, C3×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C2×C42, C3×Dic3, S3×C6, C4×Dic3, C4×C12, S3×C2×C4, C22×Dic3, C22×C12, S3×C12, C6×Dic3, S3×C2×C6, C2×C4×Dic3, C2×C4×C12, Dic3×C12, S3×C2×C12, Dic3×C2×C6, Dic3×C2×C12
►On 96 points
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 85)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)
(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)
(1 37 5 41 9 45)(2 38 6 42 10 46)(3 39 7 43 11 47)(4 40 8 44 12 48)(13 56 21 52 17 60)(14 57 22 53 18 49)(15 58 23 54 19 50)(16 59 24 55 20 51)(25 82 33 78 29 74)(26 83 34 79 30 75)(27 84 35 80 31 76)(28 73 36 81 32 77)(61 87 65 91 69 95)(62 88 66 92 70 96)(63 89 67 93 71 85)(64 90 68 94 72 86)
(1 77 41 36)(2 78 42 25)(3 79 43 26)(4 80 44 27)(5 81 45 28)(6 82 46 29)(7 83 47 30)(8 84 48 31)(9 73 37 32)(10 74 38 33)(11 75 39 34)(12 76 40 35)(13 65 52 95)(14 66 53 96)(15 67 54 85)(16 68 55 86)(17 69 56 87)(18 70 57 88)(19 71 58 89)(20 72 59 90)(21 61 60 91)(22 62 49 92)(23 63 50 93)(24 64 51 94)
G:=sub<Sym(96)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,85)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63), (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), (1,37,5,41,9,45)(2,38,6,42,10,46)(3,39,7,43,11,47)(4,40,8,44,12,48)(13,56,21,52,17,60)(14,57,22,53,18,49)(15,58,23,54,19,50)(16,59,24,55,20,51)(25,82,33,78,29,74)(26,83,34,79,30,75)(27,84,35,80,31,76)(28,73,36,81,32,77)(61,87,65,91,69,95)(62,88,66,92,70,96)(63,89,67,93,71,85)(64,90,68,94,72,86), (1,77,41,36)(2,78,42,25)(3,79,43,26)(4,80,44,27)(5,81,45,28)(6,82,46,29)(7,83,47,30)(8,84,48,31)(9,73,37,32)(10,74,38,33)(11,75,39,34)(12,76,40,35)(13,65,52,95)(14,66,53,96)(15,67,54,85)(16,68,55,86)(17,69,56,87)(18,70,57,88)(19,71,58,89)(20,72,59,90)(21,61,60,91)(22,62,49,92)(23,63,50,93)(24,64,51,94)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,85)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63), (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), (1,37,5,41,9,45)(2,38,6,42,10,46)(3,39,7,43,11,47)(4,40,8,44,12,48)(13,56,21,52,17,60)(14,57,22,53,18,49)(15,58,23,54,19,50)(16,59,24,55,20,51)(25,82,33,78,29,74)(26,83,34,79,30,75)(27,84,35,80,31,76)(28,73,36,81,32,77)(61,87,65,91,69,95)(62,88,66,92,70,96)(63,89,67,93,71,85)(64,90,68,94,72,86), (1,77,41,36)(2,78,42,25)(3,79,43,26)(4,80,44,27)(5,81,45,28)(6,82,46,29)(7,83,47,30)(8,84,48,31)(9,73,37,32)(10,74,38,33)(11,75,39,34)(12,76,40,35)(13,65,52,95)(14,66,53,96)(15,67,54,85)(16,68,55,86)(17,69,56,87)(18,70,57,88)(19,71,58,89)(20,72,59,90)(21,61,60,91)(22,62,49,92)(23,63,50,93)(24,64,51,94) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,85),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,61),(47,62),(48,63)], [(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)], [(1,37,5,41,9,45),(2,38,6,42,10,46),(3,39,7,43,11,47),(4,40,8,44,12,48),(13,56,21,52,17,60),(14,57,22,53,18,49),(15,58,23,54,19,50),(16,59,24,55,20,51),(25,82,33,78,29,74),(26,83,34,79,30,75),(27,84,35,80,31,76),(28,73,36,81,32,77),(61,87,65,91,69,95),(62,88,66,92,70,96),(63,89,67,93,71,85),(64,90,68,94,72,86)], [(1,77,41,36),(2,78,42,25),(3,79,43,26),(4,80,44,27),(5,81,45,28),(6,82,46,29),(7,83,47,30),(8,84,48,31),(9,73,37,32),(10,74,38,33),(11,75,39,34),(12,76,40,35),(13,65,52,95),(14,66,53,96),(15,67,54,85),(16,68,55,86),(17,69,56,87),(18,70,57,88),(19,71,58,89),(20,72,59,90),(21,61,60,91),(22,62,49,92),(23,63,50,93),(24,64,51,94)]])
144 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 4I | ··· | 4X | 6A | ··· | 6N | 6O | ··· | 6AI | 12A | ··· | 12P | 12Q | ··· | 12AN | 12AO | ··· | 12BT |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | D6 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C6 | S3×C12 |
| kernel | Dic3×C2×C12 | Dic3×C12 | Dic3×C2×C6 | C2×C6×C12 | C2×C4×Dic3 | C6×Dic3 | C6×C12 | C4×Dic3 | C22×Dic3 | C22×C12 | C2×Dic3 | C2×C12 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 16 | 8 | 8 | 4 | 2 | 32 | 16 | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 2 | 16 |
Matrix representation of Dic3×C2×C12 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,10],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
Dic3×C2×C12 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2\times C_{12} % in TeX
G:=Group("Dic3xC2xC12"); // GroupNames label
G:=SmallGroup(288,693);
// by ID
G=gap.SmallGroup(288,693);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,268,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations