direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.6Q8, C122.9C2, C12.33Dic6, C62.161C23, C6.3(C6×Q8), (C4×C12).9C6, C12.6(C3×Q8), (C4×C12).25S3, C4⋊Dic3.5C6, C42.5(C3×S3), (C3×C12).24Q8, C4.6(C3×Dic6), C2.5(C6×Dic6), (C2×C12).438D6, Dic3⋊C4.1C6, C6.49(C2×Dic6), C6.111(C4○D12), C32⋊9(C42.C2), (C6×C12).279C22, (C6×Dic3).88C22, C6.2(C3×C4○D4), (C2×C4).63(S3×C6), C2.6(C3×C4○D12), C22.35(S3×C2×C6), (C3×C6).46(C2×Q8), C3⋊1(C3×C42.C2), (C2×C12).86(C2×C6), (C3×C6).92(C4○D4), (C3×C4⋊Dic3).24C2, (C2×C6).16(C22×C6), (C2×Dic3).2(C2×C6), (C3×Dic3⋊C4).13C2, (C2×C6).294(C22×S3), SmallGroup(288,641)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.6Q8
G = < a,b,c,d | a3=b12=c4=1, d2=b6c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b6c-1 >
Subgroups: 242 in 127 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C42.C2, C3×Dic3, C3×C12, C3×C12, C62, Dic3⋊C4, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C6×Dic3, C6×C12, C6×C12, C12.6Q8, C3×C42.C2, C3×Dic3⋊C4, C3×C4⋊Dic3, C122, C3×C12.6Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C4○D4, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, C42.C2, S3×C6, C2×Dic6, C4○D12, C6×Q8, C3×C4○D4, C3×Dic6, S3×C2×C6, C12.6Q8, C3×C42.C2, C6×Dic6, C3×C4○D12, C3×C12.6Q8
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(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 59 67 32)(2 60 68 33)(3 49 69 34)(4 50 70 35)(5 51 71 36)(6 52 72 25)(7 53 61 26)(8 54 62 27)(9 55 63 28)(10 56 64 29)(11 57 65 30)(12 58 66 31)(13 89 43 79)(14 90 44 80)(15 91 45 81)(16 92 46 82)(17 93 47 83)(18 94 48 84)(19 95 37 73)(20 96 38 74)(21 85 39 75)(22 86 40 76)(23 87 41 77)(24 88 42 78)
(1 37 61 13)(2 48 62 24)(3 47 63 23)(4 46 64 22)(5 45 65 21)(6 44 66 20)(7 43 67 19)(8 42 68 18)(9 41 69 17)(10 40 70 16)(11 39 71 15)(12 38 72 14)(25 74 58 90)(26 73 59 89)(27 84 60 88)(28 83 49 87)(29 82 50 86)(30 81 51 85)(31 80 52 96)(32 79 53 95)(33 78 54 94)(34 77 55 93)(35 76 56 92)(36 75 57 91)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,59,67,32)(2,60,68,33)(3,49,69,34)(4,50,70,35)(5,51,71,36)(6,52,72,25)(7,53,61,26)(8,54,62,27)(9,55,63,28)(10,56,64,29)(11,57,65,30)(12,58,66,31)(13,89,43,79)(14,90,44,80)(15,91,45,81)(16,92,46,82)(17,93,47,83)(18,94,48,84)(19,95,37,73)(20,96,38,74)(21,85,39,75)(22,86,40,76)(23,87,41,77)(24,88,42,78), (1,37,61,13)(2,48,62,24)(3,47,63,23)(4,46,64,22)(5,45,65,21)(6,44,66,20)(7,43,67,19)(8,42,68,18)(9,41,69,17)(10,40,70,16)(11,39,71,15)(12,38,72,14)(25,74,58,90)(26,73,59,89)(27,84,60,88)(28,83,49,87)(29,82,50,86)(30,81,51,85)(31,80,52,96)(32,79,53,95)(33,78,54,94)(34,77,55,93)(35,76,56,92)(36,75,57,91)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,59,67,32)(2,60,68,33)(3,49,69,34)(4,50,70,35)(5,51,71,36)(6,52,72,25)(7,53,61,26)(8,54,62,27)(9,55,63,28)(10,56,64,29)(11,57,65,30)(12,58,66,31)(13,89,43,79)(14,90,44,80)(15,91,45,81)(16,92,46,82)(17,93,47,83)(18,94,48,84)(19,95,37,73)(20,96,38,74)(21,85,39,75)(22,86,40,76)(23,87,41,77)(24,88,42,78), (1,37,61,13)(2,48,62,24)(3,47,63,23)(4,46,64,22)(5,45,65,21)(6,44,66,20)(7,43,67,19)(8,42,68,18)(9,41,69,17)(10,40,70,16)(11,39,71,15)(12,38,72,14)(25,74,58,90)(26,73,59,89)(27,84,60,88)(28,83,49,87)(29,82,50,86)(30,81,51,85)(31,80,52,96)(32,79,53,95)(33,78,54,94)(34,77,55,93)(35,76,56,92)(36,75,57,91) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(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)], [(1,59,67,32),(2,60,68,33),(3,49,69,34),(4,50,70,35),(5,51,71,36),(6,52,72,25),(7,53,61,26),(8,54,62,27),(9,55,63,28),(10,56,64,29),(11,57,65,30),(12,58,66,31),(13,89,43,79),(14,90,44,80),(15,91,45,81),(16,92,46,82),(17,93,47,83),(18,94,48,84),(19,95,37,73),(20,96,38,74),(21,85,39,75),(22,86,40,76),(23,87,41,77),(24,88,42,78)], [(1,37,61,13),(2,48,62,24),(3,47,63,23),(4,46,64,22),(5,45,65,21),(6,44,66,20),(7,43,67,19),(8,42,68,18),(9,41,69,17),(10,40,70,16),(11,39,71,15),(12,38,72,14),(25,74,58,90),(26,73,59,89),(27,84,60,88),(28,83,49,87),(29,82,50,86),(30,81,51,85),(31,80,52,96),(32,79,53,95),(33,78,54,94),(34,77,55,93),(35,76,56,92),(36,75,57,91)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12AV | 12AW | ··· | 12BD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | C4○D4 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | C4○D12 | C3×C4○D4 | C3×Dic6 | C3×C4○D12 |
| kernel | C3×C12.6Q8 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C122 | C12.6Q8 | Dic3⋊C4 | C4⋊Dic3 | C4×C12 | C4×C12 | C3×C12 | C2×C12 | C3×C6 | C42 | C12 | C12 | C2×C4 | C6 | C6 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 1 | 2 | 3 | 4 | 2 | 4 | 4 | 6 | 8 | 8 | 8 | 16 |
Matrix representation of C3×C12.6Q8 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 11 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 7 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 10 | 8 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[11,0,0,0,0,6,0,0,0,0,3,7,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,5,10,0,0,0,8],[0,12,0,0,1,0,0,0,0,0,5,0,0,0,8,8] >;
C3×C12.6Q8 in GAP, Magma, Sage, TeX
C_3\times C_{12}._6Q_8 % in TeX
G:=Group("C3xC12.6Q8"); // GroupNames label
G:=SmallGroup(288,641);
// by ID
G=gap.SmallGroup(288,641);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,176,590,268,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=b^6*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations