direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C24, C24⋊4C12, C3⋊C8⋊5C12, C3⋊2(C4×C24), C32⋊8(C4×C8), (C3×C24)⋊10C4, C6.24(S3×C8), C2.2(S3×C24), C6.3(C4×C12), C6.3(C2×C24), C4.20(S3×C12), (C2×C24).23C6, (C6×C24).19C2, (C2×C24).38S3, C12.111(C4×S3), C12.25(C2×C12), (C2×C12).453D6, C62.67(C2×C4), C22.8(S3×C12), (C3×C6).13C42, C2.2(Dic3×C12), C4.11(C6×Dic3), C6.19(C4×Dic3), (C6×Dic3).15C4, (C2×Dic3).7C12, (C4×Dic3).10C6, C12.68(C2×Dic3), (C6×C12).331C22, (Dic3×C12).21C2, (C3×C3⋊C8)⋊11C4, (C6×C3⋊C8).24C2, (C2×C3⋊C8).12C6, (C2×C8).10(C3×S3), (C3×C6).29(C2×C8), (C2×C6).78(C4×S3), (C2×C4).90(S3×C6), (C2×C6).12(C2×C12), (C2×C12).120(C2×C6), (C3×C12).105(C2×C4), SmallGroup(288,247)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C24 |
Generators and relations for Dic3×C24
G = < a,b,c | a24=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 154 in 99 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C3×C6, C3×C6, C3⋊C8, C24, C24, C2×Dic3, C2×C12, C2×C12, C4×C8, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C2×C24, C3×C3⋊C8, C3×C24, C6×Dic3, C6×C12, C8×Dic3, C4×C24, C6×C3⋊C8, Dic3×C12, C6×C24, Dic3×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C42, C2×C8, C3×S3, C24, C4×S3, C2×Dic3, C2×C12, C4×C8, C3×Dic3, S3×C6, S3×C8, C4×Dic3, C4×C12, C2×C24, S3×C12, C6×Dic3, C8×Dic3, C4×C24, S3×C24, Dic3×C12, Dic3×C24
►On 96 points
(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 28 9 36 17 44)(2 29 10 37 18 45)(3 30 11 38 19 46)(4 31 12 39 20 47)(5 32 13 40 21 48)(6 33 14 41 22 25)(7 34 15 42 23 26)(8 35 16 43 24 27)(49 96 65 88 57 80)(50 73 66 89 58 81)(51 74 67 90 59 82)(52 75 68 91 60 83)(53 76 69 92 61 84)(54 77 70 93 62 85)(55 78 71 94 63 86)(56 79 72 95 64 87)
(1 96 36 57)(2 73 37 58)(3 74 38 59)(4 75 39 60)(5 76 40 61)(6 77 41 62)(7 78 42 63)(8 79 43 64)(9 80 44 65)(10 81 45 66)(11 82 46 67)(12 83 47 68)(13 84 48 69)(14 85 25 70)(15 86 26 71)(16 87 27 72)(17 88 28 49)(18 89 29 50)(19 90 30 51)(20 91 31 52)(21 92 32 53)(22 93 33 54)(23 94 34 55)(24 95 35 56)
G:=sub<Sym(96)| (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), (1,28,9,36,17,44)(2,29,10,37,18,45)(3,30,11,38,19,46)(4,31,12,39,20,47)(5,32,13,40,21,48)(6,33,14,41,22,25)(7,34,15,42,23,26)(8,35,16,43,24,27)(49,96,65,88,57,80)(50,73,66,89,58,81)(51,74,67,90,59,82)(52,75,68,91,60,83)(53,76,69,92,61,84)(54,77,70,93,62,85)(55,78,71,94,63,86)(56,79,72,95,64,87), (1,96,36,57)(2,73,37,58)(3,74,38,59)(4,75,39,60)(5,76,40,61)(6,77,41,62)(7,78,42,63)(8,79,43,64)(9,80,44,65)(10,81,45,66)(11,82,46,67)(12,83,47,68)(13,84,48,69)(14,85,25,70)(15,86,26,71)(16,87,27,72)(17,88,28,49)(18,89,29,50)(19,90,30,51)(20,91,31,52)(21,92,32,53)(22,93,33,54)(23,94,34,55)(24,95,35,56)>;
G:=Group( (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), (1,28,9,36,17,44)(2,29,10,37,18,45)(3,30,11,38,19,46)(4,31,12,39,20,47)(5,32,13,40,21,48)(6,33,14,41,22,25)(7,34,15,42,23,26)(8,35,16,43,24,27)(49,96,65,88,57,80)(50,73,66,89,58,81)(51,74,67,90,59,82)(52,75,68,91,60,83)(53,76,69,92,61,84)(54,77,70,93,62,85)(55,78,71,94,63,86)(56,79,72,95,64,87), (1,96,36,57)(2,73,37,58)(3,74,38,59)(4,75,39,60)(5,76,40,61)(6,77,41,62)(7,78,42,63)(8,79,43,64)(9,80,44,65)(10,81,45,66)(11,82,46,67)(12,83,47,68)(13,84,48,69)(14,85,25,70)(15,86,26,71)(16,87,27,72)(17,88,28,49)(18,89,29,50)(19,90,30,51)(20,91,31,52)(21,92,32,53)(22,93,33,54)(23,94,34,55)(24,95,35,56) );
G=PermutationGroup([[(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)], [(1,28,9,36,17,44),(2,29,10,37,18,45),(3,30,11,38,19,46),(4,31,12,39,20,47),(5,32,13,40,21,48),(6,33,14,41,22,25),(7,34,15,42,23,26),(8,35,16,43,24,27),(49,96,65,88,57,80),(50,73,66,89,58,81),(51,74,67,90,59,82),(52,75,68,91,60,83),(53,76,69,92,61,84),(54,77,70,93,62,85),(55,78,71,94,63,86),(56,79,72,95,64,87)], [(1,96,36,57),(2,73,37,58),(3,74,38,59),(4,75,39,60),(5,76,40,61),(6,77,41,62),(7,78,42,63),(8,79,43,64),(9,80,44,65),(10,81,45,66),(11,82,46,67),(12,83,47,68),(13,84,48,69),(14,85,25,70),(15,86,26,71),(16,87,27,72),(17,88,28,49),(18,89,29,50),(19,90,30,51),(20,91,31,52),(21,92,32,53),(22,93,33,54),(23,94,34,55),(24,95,35,56)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6O | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AJ | 24A | ··· | 24P | 24Q | ··· | 24AN | 24AO | ··· | 24BD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 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 |
| type | + | + | + | + | + | - | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C24 | S3 | Dic3 | D6 | C3×S3 | C4×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C8 | S3×C12 | S3×C12 | S3×C24 |
| kernel | Dic3×C24 | C6×C3⋊C8 | Dic3×C12 | C6×C24 | C8×Dic3 | C3×C3⋊C8 | C3×C24 | C6×Dic3 | C2×C3⋊C8 | C4×Dic3 | C2×C24 | C3×Dic3 | C3⋊C8 | C24 | C2×Dic3 | Dic3 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 16 | 8 | 8 | 8 | 32 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of Dic3×C24 ►in GL4(𝔽73) generated by
| 7 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 19 | 8 |
| 72 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 6 | 63 |
| 0 | 0 | 40 | 67 |
G:=sub<GL(4,GF(73))| [7,0,0,0,0,52,0,0,0,0,49,0,0,0,0,49],[1,0,0,0,0,72,0,0,0,0,64,19,0,0,0,8],[72,0,0,0,0,46,0,0,0,0,6,40,0,0,63,67] >;
Dic3×C24 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{24} % in TeX
G:=Group("Dic3xC24"); // GroupNames label
G:=SmallGroup(288,247);
// by ID
G=gap.SmallGroup(288,247);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,176,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations