metabelian, supersoluble, monomial
Aliases: C6.16D24, D12⋊1Dic3, C62.18D4, (C3×D12)⋊2C4, (C3×C6).14D8, C12.14(C4×S3), C6.5(D4⋊S3), (C2×D12).1S3, (C6×D12).3C2, (C2×C12).76D6, (C3×C12).33D4, (C2×C6).49D12, (C3×C6).8SD16, C4.1(S3×Dic3), C6.35(D6⋊C4), C3⋊3(C2.D24), C6.1(D4.S3), C6.10(C24⋊C2), C3⋊1(D4⋊Dic3), C2.1(C3⋊D24), C12.29(C3⋊D4), C32⋊5(D4⋊C4), C2.6(D6⋊Dic3), (C6×C12).25C22, C12.22(C2×Dic3), C12⋊Dic3⋊13C2, C4.11(D6⋊S3), C6.5(C6.D4), C2.1(D12.S3), C22.12(C3⋊D12), (C6×C3⋊C8)⋊3C2, (C2×C3⋊C8)⋊1S3, (C2×C4).52S32, (C3×C12).23(C2×C4), (C2×C6).30(C3⋊D4), (C3×C6).30(C22⋊C4), SmallGroup(288,211)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.16D24
G = < a,b,c | a6=b24=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >
Subgroups: 458 in 113 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3×C6, C3⋊C8, C24, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, C3⋊Dic3, C3×C12, S3×C6, C62, C2×C3⋊C8, C4⋊Dic3, C2×C24, C2×D12, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C2.D24, D4⋊Dic3, C6×C3⋊C8, C12⋊Dic3, C6×D12, C6.16D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C4×S3, D12, C2×Dic3, C3⋊D4, D4⋊C4, S32, C24⋊C2, D24, D6⋊C4, D4⋊S3, D4.S3, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C2.D24, D4⋊Dic3, C3⋊D24, D12.S3, D6⋊Dic3, C6.16D24
►On 96 points
(1 71 9 55 17 63)(2 64 18 56 10 72)(3 49 11 57 19 65)(4 66 20 58 12 50)(5 51 13 59 21 67)(6 68 22 60 14 52)(7 53 15 61 23 69)(8 70 24 62 16 54)(25 90 41 82 33 74)(26 75 34 83 42 91)(27 92 43 84 35 76)(28 77 36 85 44 93)(29 94 45 86 37 78)(30 79 38 87 46 95)(31 96 47 88 39 80)(32 81 40 89 48 73)
(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 55 77)(2 76 56 43)(3 42 57 75)(4 74 58 41)(5 40 59 73)(6 96 60 39)(7 38 61 95)(8 94 62 37)(9 36 63 93)(10 92 64 35)(11 34 65 91)(12 90 66 33)(13 32 67 89)(14 88 68 31)(15 30 69 87)(16 86 70 29)(17 28 71 85)(18 84 72 27)(19 26 49 83)(20 82 50 25)(21 48 51 81)(22 80 52 47)(23 46 53 79)(24 78 54 45)
G:=sub<Sym(96)| (1,71,9,55,17,63)(2,64,18,56,10,72)(3,49,11,57,19,65)(4,66,20,58,12,50)(5,51,13,59,21,67)(6,68,22,60,14,52)(7,53,15,61,23,69)(8,70,24,62,16,54)(25,90,41,82,33,74)(26,75,34,83,42,91)(27,92,43,84,35,76)(28,77,36,85,44,93)(29,94,45,86,37,78)(30,79,38,87,46,95)(31,96,47,88,39,80)(32,81,40,89,48,73), (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), (1,44,55,77)(2,76,56,43)(3,42,57,75)(4,74,58,41)(5,40,59,73)(6,96,60,39)(7,38,61,95)(8,94,62,37)(9,36,63,93)(10,92,64,35)(11,34,65,91)(12,90,66,33)(13,32,67,89)(14,88,68,31)(15,30,69,87)(16,86,70,29)(17,28,71,85)(18,84,72,27)(19,26,49,83)(20,82,50,25)(21,48,51,81)(22,80,52,47)(23,46,53,79)(24,78,54,45)>;
G:=Group( (1,71,9,55,17,63)(2,64,18,56,10,72)(3,49,11,57,19,65)(4,66,20,58,12,50)(5,51,13,59,21,67)(6,68,22,60,14,52)(7,53,15,61,23,69)(8,70,24,62,16,54)(25,90,41,82,33,74)(26,75,34,83,42,91)(27,92,43,84,35,76)(28,77,36,85,44,93)(29,94,45,86,37,78)(30,79,38,87,46,95)(31,96,47,88,39,80)(32,81,40,89,48,73), (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), (1,44,55,77)(2,76,56,43)(3,42,57,75)(4,74,58,41)(5,40,59,73)(6,96,60,39)(7,38,61,95)(8,94,62,37)(9,36,63,93)(10,92,64,35)(11,34,65,91)(12,90,66,33)(13,32,67,89)(14,88,68,31)(15,30,69,87)(16,86,70,29)(17,28,71,85)(18,84,72,27)(19,26,49,83)(20,82,50,25)(21,48,51,81)(22,80,52,47)(23,46,53,79)(24,78,54,45) );
G=PermutationGroup([[(1,71,9,55,17,63),(2,64,18,56,10,72),(3,49,11,57,19,65),(4,66,20,58,12,50),(5,51,13,59,21,67),(6,68,22,60,14,52),(7,53,15,61,23,69),(8,70,24,62,16,54),(25,90,41,82,33,74),(26,75,34,83,42,91),(27,92,43,84,35,76),(28,77,36,85,44,93),(29,94,45,86,37,78),(30,79,38,87,46,95),(31,96,47,88,39,80),(32,81,40,89,48,73)], [(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)], [(1,44,55,77),(2,76,56,43),(3,42,57,75),(4,74,58,41),(5,40,59,73),(6,96,60,39),(7,38,61,95),(8,94,62,37),(9,36,63,93),(10,92,64,35),(11,34,65,91),(12,90,66,33),(13,32,67,89),(14,88,68,31),(15,30,69,87),(16,86,70,29),(17,28,71,85),(18,84,72,27),(19,26,49,83),(20,82,50,25),(21,48,51,81),(22,80,52,47),(23,46,53,79),(24,78,54,45)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | D4 | D4 | Dic3 | D6 | D8 | SD16 | C4×S3 | C3⋊D4 | D12 | C3⋊D4 | C24⋊C2 | D24 | S32 | D4⋊S3 | D4.S3 | S3×Dic3 | D6⋊S3 | C3⋊D12 | C3⋊D24 | D12.S3 |
| kernel | C6.16D24 | C6×C3⋊C8 | C12⋊Dic3 | C6×D12 | C3×D12 | C2×C3⋊C8 | C2×D12 | C3×C12 | C62 | D12 | 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 | 2 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C6.16D24 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 10 | 35 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 66 | 20 | 0 | 0 | 0 | 0 |
| 56 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 0 | 27 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[10,0,0,0,0,0,35,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,30,0,0,0,0,43,30],[66,56,0,0,0,0,20,7,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27] >;
C6.16D24 in GAP, Magma, Sage, TeX
C_6._{16}D_{24} % in TeX
G:=Group("C6.16D24"); // GroupNames label
G:=SmallGroup(288,211);
// by ID
G=gap.SmallGroup(288,211);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,422,100,1356,9414]);
// 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=a^3*b^-1>;
// generators/relations