metabelian, supersoluble, monomial
Aliases: C6.18D24, C12.7Dic6, C6.8Dic12, C62.28D4, C3⋊C8⋊2Dic3, (C3×C6).16D8, (C3×C12).7Q8, (C3×C6).6Q16, C12.90(C4×S3), C6.7(D4⋊S3), (C2×C12).82D6, C3⋊1(C24⋊1C4), (C2×C6).55D12, C4⋊Dic3.3S3, C32⋊4(C2.D8), C3⋊1(C6.Q16), C6.3(C4⋊Dic3), C4.12(S3×Dic3), C6.3(C3⋊Q16), C2.3(C3⋊D24), C6.5(Dic3⋊C4), (C6×C12).35C22, C12.11(C2×Dic3), C4.2(C32⋊2Q8), C2.3(C32⋊3Q16), C2.4(Dic3⋊Dic3), C12⋊Dic3.11C2, C22.17(C3⋊D12), (C3×C3⋊C8)⋊1C4, (C6×C3⋊C8).9C2, (C2×C4).59S32, (C2×C3⋊C8).4S3, (C3×C6).22(C4⋊C4), (C3×C12).33(C2×C4), (C3×C4⋊Dic3).5C2, (C2×C6).35(C3⋊D4), SmallGroup(288,223)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.18D24
G = < a,b,c | a6=b24=1, c2=a3, bab-1=cac-1=a-1, cbc-1=b-1 >
Subgroups: 298 in 85 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C2.D8, C3×Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C3×C3⋊C8, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6.Q16, C24⋊1C4, C6×C3⋊C8, C3×C4⋊Dic3, C12⋊Dic3, C6.18D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D8, Q16, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.D8, S32, D24, Dic12, Dic3⋊C4, C4⋊Dic3, D4⋊S3, C3⋊Q16, S3×Dic3, C3⋊D12, C32⋊2Q8, C6.Q16, C24⋊1C4, C3⋊D24, C32⋊3Q16, Dic3⋊Dic3, C6.18D24
►On 96 points
(1 40 9 48 17 32)(2 33 18 25 10 41)(3 42 11 26 19 34)(4 35 20 27 12 43)(5 44 13 28 21 36)(6 37 22 29 14 45)(7 46 15 30 23 38)(8 39 24 31 16 47)(49 88 57 96 65 80)(50 81 66 73 58 89)(51 90 59 74 67 82)(52 83 68 75 60 91)(53 92 61 76 69 84)(54 85 70 77 62 93)(55 94 63 78 71 86)(56 87 72 79 64 95)
(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 69 48 92)(2 68 25 91)(3 67 26 90)(4 66 27 89)(5 65 28 88)(6 64 29 87)(7 63 30 86)(8 62 31 85)(9 61 32 84)(10 60 33 83)(11 59 34 82)(12 58 35 81)(13 57 36 80)(14 56 37 79)(15 55 38 78)(16 54 39 77)(17 53 40 76)(18 52 41 75)(19 51 42 74)(20 50 43 73)(21 49 44 96)(22 72 45 95)(23 71 46 94)(24 70 47 93)
G:=sub<Sym(96)| (1,40,9,48,17,32)(2,33,18,25,10,41)(3,42,11,26,19,34)(4,35,20,27,12,43)(5,44,13,28,21,36)(6,37,22,29,14,45)(7,46,15,30,23,38)(8,39,24,31,16,47)(49,88,57,96,65,80)(50,81,66,73,58,89)(51,90,59,74,67,82)(52,83,68,75,60,91)(53,92,61,76,69,84)(54,85,70,77,62,93)(55,94,63,78,71,86)(56,87,72,79,64,95), (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), (1,69,48,92)(2,68,25,91)(3,67,26,90)(4,66,27,89)(5,65,28,88)(6,64,29,87)(7,63,30,86)(8,62,31,85)(9,61,32,84)(10,60,33,83)(11,59,34,82)(12,58,35,81)(13,57,36,80)(14,56,37,79)(15,55,38,78)(16,54,39,77)(17,53,40,76)(18,52,41,75)(19,51,42,74)(20,50,43,73)(21,49,44,96)(22,72,45,95)(23,71,46,94)(24,70,47,93)>;
G:=Group( (1,40,9,48,17,32)(2,33,18,25,10,41)(3,42,11,26,19,34)(4,35,20,27,12,43)(5,44,13,28,21,36)(6,37,22,29,14,45)(7,46,15,30,23,38)(8,39,24,31,16,47)(49,88,57,96,65,80)(50,81,66,73,58,89)(51,90,59,74,67,82)(52,83,68,75,60,91)(53,92,61,76,69,84)(54,85,70,77,62,93)(55,94,63,78,71,86)(56,87,72,79,64,95), (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), (1,69,48,92)(2,68,25,91)(3,67,26,90)(4,66,27,89)(5,65,28,88)(6,64,29,87)(7,63,30,86)(8,62,31,85)(9,61,32,84)(10,60,33,83)(11,59,34,82)(12,58,35,81)(13,57,36,80)(14,56,37,79)(15,55,38,78)(16,54,39,77)(17,53,40,76)(18,52,41,75)(19,51,42,74)(20,50,43,73)(21,49,44,96)(22,72,45,95)(23,71,46,94)(24,70,47,93) );
G=PermutationGroup([[(1,40,9,48,17,32),(2,33,18,25,10,41),(3,42,11,26,19,34),(4,35,20,27,12,43),(5,44,13,28,21,36),(6,37,22,29,14,45),(7,46,15,30,23,38),(8,39,24,31,16,47),(49,88,57,96,65,80),(50,81,66,73,58,89),(51,90,59,74,67,82),(52,83,68,75,60,91),(53,92,61,76,69,84),(54,85,70,77,62,93),(55,94,63,78,71,86),(56,87,72,79,64,95)], [(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)], [(1,69,48,92),(2,68,25,91),(3,67,26,90),(4,66,27,89),(5,65,28,88),(6,64,29,87),(7,63,30,86),(8,62,31,85),(9,61,32,84),(10,60,33,83),(11,59,34,82),(12,58,35,81),(13,57,36,80),(14,56,37,79),(15,55,38,78),(16,54,39,77),(17,53,40,76),(18,52,41,75),(19,51,42,74),(20,50,43,73),(21,49,44,96),(22,72,45,95),(23,71,46,94),(24,70,47,93)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | - | + | + | - | + | + | - | - | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | Q8 | D4 | Dic3 | D6 | D8 | Q16 | Dic6 | C4×S3 | D12 | C3⋊D4 | D24 | Dic12 | S32 | D4⋊S3 | C3⋊Q16 | S3×Dic3 | C32⋊2Q8 | C3⋊D12 | C3⋊D24 | C32⋊3Q16 |
| kernel | C6.18D24 | C6×C3⋊C8 | C3×C4⋊Dic3 | C12⋊Dic3 | C3×C3⋊C8 | C2×C3⋊C8 | C4⋊Dic3 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C6 | C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C6.18D24 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 63 | 0 | 0 | 0 | 0 | 0 |
| 59 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 0 | 0 | 0 | 0 | 66 | 66 |
| 55 | 68 | 0 | 0 | 0 | 0 |
| 65 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[63,59,0,0,0,0,0,51,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,59,66,0,0,0,0,7,66],[55,65,0,0,0,0,68,18,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;
C6.18D24 in GAP, Magma, Sage, TeX
C_6._{18}D_{24} % in TeX
G:=Group("C6.18D24"); // GroupNames label
G:=SmallGroup(288,223);
// by ID
G=gap.SmallGroup(288,223);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,148,422,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations