direct product, metabelian, supersoluble, monomial
Aliases: C6×D24, C24⋊25D6, C12.68D12, C62.86D4, C8⋊7(S3×C6), C6⋊1(C3×D8), C3⋊1(C6×D8), (C3×C6)⋊4D8, C24⋊8(C2×C6), (C2×C24)⋊5C6, (C2×C24)⋊8S3, C32⋊9(C2×D8), (C6×C24)⋊10C2, D12⋊3(C2×C6), (C2×D12)⋊4C6, C4.7(C3×D12), C6.10(C6×D4), (C6×D12)⋊28C2, C6.98(C2×D12), C2.12(C6×D12), (C2×C6).74D12, C12.30(C3×D4), (C3×C24)⋊25C22, (C3×C12).132D4, (C2×C12).442D6, (C3×D12)⋊38C22, C12.29(C22×C6), C22.13(C3×D12), C12.216(C22×S3), (C3×C12).161C23, (C6×C12).322C22, (C2×C8)⋊3(C3×S3), C4.27(S3×C2×C6), (C2×C4).79(S3×C6), (C2×C6).21(C3×D4), (C3×C6).180(C2×D4), (C2×C12).107(C2×C6), SmallGroup(288,674)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D24
G = < a,b,c | a6=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 538 in 163 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, C2×D4, C3×S3, C3×C6, C3×C6, C24, C24, D12, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×D8, C3×C12, S3×C6, C62, D24, C2×C24, C2×C24, C3×D8, C2×D12, C6×D4, C3×C24, C3×D12, C3×D12, C6×C12, S3×C2×C6, C2×D24, C6×D8, C3×D24, C6×C24, C6×D12, C6×D24
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, D8, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C2×D8, S3×C6, D24, C3×D8, C2×D12, C6×D4, C3×D12, S3×C2×C6, C2×D24, C6×D8, C3×D24, C6×D12, C6×D24
►On 96 points
(1 49 9 57 17 65)(2 50 10 58 18 66)(3 51 11 59 19 67)(4 52 12 60 20 68)(5 53 13 61 21 69)(6 54 14 62 22 70)(7 55 15 63 23 71)(8 56 16 64 24 72)(25 90 41 82 33 74)(26 91 42 83 34 75)(27 92 43 84 35 76)(28 93 44 85 36 77)(29 94 45 86 37 78)(30 95 46 87 38 79)(31 96 47 88 39 80)(32 73 48 89 40 81)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 48)(22 47)(23 46)(24 45)(49 85)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)(61 73)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)
G:=sub<Sym(96)| (1,49,9,57,17,65)(2,50,10,58,18,66)(3,51,11,59,19,67)(4,52,12,60,20,68)(5,53,13,61,21,69)(6,54,14,62,22,70)(7,55,15,63,23,71)(8,56,16,64,24,72)(25,90,41,82,33,74)(26,91,42,83,34,75)(27,92,43,84,35,76)(28,93,44,85,36,77)(29,94,45,86,37,78)(30,95,46,87,38,79)(31,96,47,88,39,80)(32,73,48,89,40,81), (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,48)(22,47)(23,46)(24,45)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)(61,73)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)>;
G:=Group( (1,49,9,57,17,65)(2,50,10,58,18,66)(3,51,11,59,19,67)(4,52,12,60,20,68)(5,53,13,61,21,69)(6,54,14,62,22,70)(7,55,15,63,23,71)(8,56,16,64,24,72)(25,90,41,82,33,74)(26,91,42,83,34,75)(27,92,43,84,35,76)(28,93,44,85,36,77)(29,94,45,86,37,78)(30,95,46,87,38,79)(31,96,47,88,39,80)(32,73,48,89,40,81), (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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,48)(22,47)(23,46)(24,45)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)(61,73)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86) );
G=PermutationGroup([[(1,49,9,57,17,65),(2,50,10,58,18,66),(3,51,11,59,19,67),(4,52,12,60,20,68),(5,53,13,61,21,69),(6,54,14,62,22,70),(7,55,15,63,23,71),(8,56,16,64,24,72),(25,90,41,82,33,74),(26,91,42,83,34,75),(27,92,43,84,35,76),(28,93,44,85,36,77),(29,94,45,86,37,78),(30,95,46,87,38,79),(31,96,47,88,39,80),(32,73,48,89,40,81)], [(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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,48),(22,47),(23,46),(24,45),(49,85),(50,84),(51,83),(52,82),(53,81),(54,80),(55,79),(56,78),(57,77),(58,76),(59,75),(60,74),(61,73),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | D8 | C3×S3 | D12 | C3×D4 | D12 | C3×D4 | S3×C6 | S3×C6 | D24 | C3×D8 | C3×D12 | C3×D12 | C3×D24 |
| kernel | C6×D24 | C3×D24 | C6×C24 | C6×D12 | C2×D24 | D24 | C2×C24 | C2×D12 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C2×C8 | C12 | C12 | C2×C6 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C6×D24 ►in GL5(𝔽73)
| 9 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 | 0 |
| 0 | 0 | 70 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 16 | 41 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 52 |
| 0 | 0 | 0 | 36 | 63 |
G:=sub<GL(5,GF(73))| [9,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,24,0,0,0,0,0,70,0,0,0,0,0,0,16,0,0,0,41,41],[72,0,0,0,0,0,0,64,0,0,0,8,0,0,0,0,0,0,10,36,0,0,0,52,63] >;
C6×D24 in GAP, Magma, Sage, TeX
C_6\times D_{24} % in TeX
G:=Group("C6xD24"); // GroupNames label
G:=SmallGroup(288,674);
// by ID
G=gap.SmallGroup(288,674);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,394,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations