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 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×5], C12 [×4], C12 [×3], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], C2.D8, C3×Dic3, C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3 [×3], C3×C4⋊C4, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C2×C3⋊Dic3, C6×C12, C6.Q16, C24⋊1C4, C6×C3⋊C8, C3×C4⋊Dic3, C12⋊Dic3, C6.18D24
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, D8, Q16, Dic6 [×2], 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 25 9 33 17 41)(2 42 18 34 10 26)(3 27 11 35 19 43)(4 44 20 36 12 28)(5 29 13 37 21 45)(6 46 22 38 14 30)(7 31 15 39 23 47)(8 48 24 40 16 32)(49 93 65 85 57 77)(50 78 58 86 66 94)(51 95 67 87 59 79)(52 80 60 88 68 96)(53 73 69 89 61 81)(54 82 62 90 70 74)(55 75 71 91 63 83)(56 84 64 92 72 76)
(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 84 33 72)(2 83 34 71)(3 82 35 70)(4 81 36 69)(5 80 37 68)(6 79 38 67)(7 78 39 66)(8 77 40 65)(9 76 41 64)(10 75 42 63)(11 74 43 62)(12 73 44 61)(13 96 45 60)(14 95 46 59)(15 94 47 58)(16 93 48 57)(17 92 25 56)(18 91 26 55)(19 90 27 54)(20 89 28 53)(21 88 29 52)(22 87 30 51)(23 86 31 50)(24 85 32 49)
G:=sub<Sym(96)| (1,25,9,33,17,41)(2,42,18,34,10,26)(3,27,11,35,19,43)(4,44,20,36,12,28)(5,29,13,37,21,45)(6,46,22,38,14,30)(7,31,15,39,23,47)(8,48,24,40,16,32)(49,93,65,85,57,77)(50,78,58,86,66,94)(51,95,67,87,59,79)(52,80,60,88,68,96)(53,73,69,89,61,81)(54,82,62,90,70,74)(55,75,71,91,63,83)(56,84,64,92,72,76), (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), (1,84,33,72)(2,83,34,71)(3,82,35,70)(4,81,36,69)(5,80,37,68)(6,79,38,67)(7,78,39,66)(8,77,40,65)(9,76,41,64)(10,75,42,63)(11,74,43,62)(12,73,44,61)(13,96,45,60)(14,95,46,59)(15,94,47,58)(16,93,48,57)(17,92,25,56)(18,91,26,55)(19,90,27,54)(20,89,28,53)(21,88,29,52)(22,87,30,51)(23,86,31,50)(24,85,32,49)>;
G:=Group( (1,25,9,33,17,41)(2,42,18,34,10,26)(3,27,11,35,19,43)(4,44,20,36,12,28)(5,29,13,37,21,45)(6,46,22,38,14,30)(7,31,15,39,23,47)(8,48,24,40,16,32)(49,93,65,85,57,77)(50,78,58,86,66,94)(51,95,67,87,59,79)(52,80,60,88,68,96)(53,73,69,89,61,81)(54,82,62,90,70,74)(55,75,71,91,63,83)(56,84,64,92,72,76), (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), (1,84,33,72)(2,83,34,71)(3,82,35,70)(4,81,36,69)(5,80,37,68)(6,79,38,67)(7,78,39,66)(8,77,40,65)(9,76,41,64)(10,75,42,63)(11,74,43,62)(12,73,44,61)(13,96,45,60)(14,95,46,59)(15,94,47,58)(16,93,48,57)(17,92,25,56)(18,91,26,55)(19,90,27,54)(20,89,28,53)(21,88,29,52)(22,87,30,51)(23,86,31,50)(24,85,32,49) );
G=PermutationGroup([(1,25,9,33,17,41),(2,42,18,34,10,26),(3,27,11,35,19,43),(4,44,20,36,12,28),(5,29,13,37,21,45),(6,46,22,38,14,30),(7,31,15,39,23,47),(8,48,24,40,16,32),(49,93,65,85,57,77),(50,78,58,86,66,94),(51,95,67,87,59,79),(52,80,60,88,68,96),(53,73,69,89,61,81),(54,82,62,90,70,74),(55,75,71,91,63,83),(56,84,64,92,72,76)], [(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)], [(1,84,33,72),(2,83,34,71),(3,82,35,70),(4,81,36,69),(5,80,37,68),(6,79,38,67),(7,78,39,66),(8,77,40,65),(9,76,41,64),(10,75,42,63),(11,74,43,62),(12,73,44,61),(13,96,45,60),(14,95,46,59),(15,94,47,58),(16,93,48,57),(17,92,25,56),(18,91,26,55),(19,90,27,54),(20,89,28,53),(21,88,29,52),(22,87,30,51),(23,86,31,50),(24,85,32,49)])
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