direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×C3⋊C16, C6⋊C48, C24.5C12, C12.3C24, C24.99D6, C62.6C8, C24.16Dic3, (C3×C6)⋊3C16, C3⋊2(C2×C48), C8.21(S3×C6), (C2×C6).5C24, C6.8(C2×C24), C32⋊9(C2×C16), C12.11(C3⋊C8), (C2×C24).37S3, (C3×C12).11C8, (C3×C24).10C4, (C6×C12).29C4, C24.34(C2×C6), (C2×C24).21C6, (C6×C24).18C2, C4.9(C6×Dic3), C8.4(C3×Dic3), (C2×C12).16C12, C12.45(C2×C12), (C3×C24).66C22, C12.66(C2×Dic3), (C2×C12).31Dic3, C4.3(C3×C3⋊C8), C2.2(C6×C3⋊C8), C6.19(C2×C3⋊C8), (C2×C6).9(C3⋊C8), (C2×C8).9(C3×S3), C22.2(C3×C3⋊C8), (C3×C6).39(C2×C8), (C2×C4).8(C3×Dic3), (C3×C12).130(C2×C4), SmallGroup(288,245)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C6×C3⋊C16 |
Generators and relations for C6×C3⋊C16
G = < a,b,c | a6=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 90 in 67 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C16, C2×C8, C3×C6, C3×C6, C24, C24, C2×C12, C2×C12, C2×C16, C3×C12, C62, C3⋊C16, C48, C2×C24, C2×C24, C3×C24, C6×C12, C2×C3⋊C16, C2×C48, C3×C3⋊C16, C6×C24, C6×C3⋊C16
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C16, C2×C8, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C16, C3×Dic3, S3×C6, C3⋊C16, C48, C2×C3⋊C8, C2×C24, C3×C3⋊C8, C6×Dic3, C2×C3⋊C16, C2×C48, C3×C3⋊C16, C6×C3⋊C8, C6×C3⋊C16
►On 96 points
(1 24 75 62 33 82)(2 25 76 63 34 83)(3 26 77 64 35 84)(4 27 78 49 36 85)(5 28 79 50 37 86)(6 29 80 51 38 87)(7 30 65 52 39 88)(8 31 66 53 40 89)(9 32 67 54 41 90)(10 17 68 55 42 91)(11 18 69 56 43 92)(12 19 70 57 44 93)(13 20 71 58 45 94)(14 21 72 59 46 95)(15 22 73 60 47 96)(16 23 74 61 48 81)
(1 75 33)(2 34 76)(3 77 35)(4 36 78)(5 79 37)(6 38 80)(7 65 39)(8 40 66)(9 67 41)(10 42 68)(11 69 43)(12 44 70)(13 71 45)(14 46 72)(15 73 47)(16 48 74)(17 91 55)(18 56 92)(19 93 57)(20 58 94)(21 95 59)(22 60 96)(23 81 61)(24 62 82)(25 83 63)(26 64 84)(27 85 49)(28 50 86)(29 87 51)(30 52 88)(31 89 53)(32 54 90)
(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)
G:=sub<Sym(96)| (1,24,75,62,33,82)(2,25,76,63,34,83)(3,26,77,64,35,84)(4,27,78,49,36,85)(5,28,79,50,37,86)(6,29,80,51,38,87)(7,30,65,52,39,88)(8,31,66,53,40,89)(9,32,67,54,41,90)(10,17,68,55,42,91)(11,18,69,56,43,92)(12,19,70,57,44,93)(13,20,71,58,45,94)(14,21,72,59,46,95)(15,22,73,60,47,96)(16,23,74,61,48,81), (1,75,33)(2,34,76)(3,77,35)(4,36,78)(5,79,37)(6,38,80)(7,65,39)(8,40,66)(9,67,41)(10,42,68)(11,69,43)(12,44,70)(13,71,45)(14,46,72)(15,73,47)(16,48,74)(17,91,55)(18,56,92)(19,93,57)(20,58,94)(21,95,59)(22,60,96)(23,81,61)(24,62,82)(25,83,63)(26,64,84)(27,85,49)(28,50,86)(29,87,51)(30,52,88)(31,89,53)(32,54,90), (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)>;
G:=Group( (1,24,75,62,33,82)(2,25,76,63,34,83)(3,26,77,64,35,84)(4,27,78,49,36,85)(5,28,79,50,37,86)(6,29,80,51,38,87)(7,30,65,52,39,88)(8,31,66,53,40,89)(9,32,67,54,41,90)(10,17,68,55,42,91)(11,18,69,56,43,92)(12,19,70,57,44,93)(13,20,71,58,45,94)(14,21,72,59,46,95)(15,22,73,60,47,96)(16,23,74,61,48,81), (1,75,33)(2,34,76)(3,77,35)(4,36,78)(5,79,37)(6,38,80)(7,65,39)(8,40,66)(9,67,41)(10,42,68)(11,69,43)(12,44,70)(13,71,45)(14,46,72)(15,73,47)(16,48,74)(17,91,55)(18,56,92)(19,93,57)(20,58,94)(21,95,59)(22,60,96)(23,81,61)(24,62,82)(25,83,63)(26,64,84)(27,85,49)(28,50,86)(29,87,51)(30,52,88)(31,89,53)(32,54,90), (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) );
G=PermutationGroup([[(1,24,75,62,33,82),(2,25,76,63,34,83),(3,26,77,64,35,84),(4,27,78,49,36,85),(5,28,79,50,37,86),(6,29,80,51,38,87),(7,30,65,52,39,88),(8,31,66,53,40,89),(9,32,67,54,41,90),(10,17,68,55,42,91),(11,18,69,56,43,92),(12,19,70,57,44,93),(13,20,71,58,45,94),(14,21,72,59,46,95),(15,22,73,60,47,96),(16,23,74,61,48,81)], [(1,75,33),(2,34,76),(3,77,35),(4,36,78),(5,79,37),(6,38,80),(7,65,39),(8,40,66),(9,67,41),(10,42,68),(11,69,43),(12,44,70),(13,71,45),(14,46,72),(15,73,47),(16,48,74),(17,91,55),(18,56,92),(19,93,57),(20,58,94),(21,95,59),(22,60,96),(23,81,61),(24,62,82),(25,83,63),(26,64,84),(27,85,49),(28,50,86),(29,87,51),(30,52,88),(31,89,53),(32,54,90)], [(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)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12T | 16A | ··· | 16P | 24A | ··· | 24P | 24Q | ··· | 24AN | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C16 | C24 | C24 | C48 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | C3⋊C16 | C3×C3⋊C8 | C3×C3⋊C8 | C3×C3⋊C16 |
| kernel | C6×C3⋊C16 | C3×C3⋊C16 | C6×C24 | C2×C3⋊C16 | C3×C24 | C6×C12 | C3⋊C16 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 | C2×C24 | C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 16 | 8 | 8 | 32 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C6×C3⋊C16 ►in GL3(𝔽97) generated by
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 43 | 61 |
| 96 | 0 | 0 |
| 0 | 47 | 60 |
| 0 | 9 | 50 |
G:=sub<GL(3,GF(97))| [36,0,0,0,36,0,0,0,36],[1,0,0,0,35,43,0,0,61],[96,0,0,0,47,9,0,60,50] >;
C6×C3⋊C16 in GAP, Magma, Sage, TeX
C_6\times C_3\rtimes C_{16} % in TeX
G:=Group("C6xC3:C16"); // GroupNames label
G:=SmallGroup(288,245);
// by ID
G=gap.SmallGroup(288,245);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,80,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations