metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.18D4, C8.20D12, D12.20D4, Dic6.20D4, M4(2).9D6, (C2×C8).70D6, C8.C4⋊5S3, C8○D12.2C2, C4.135(S3×D4), C4.56(C2×D12), C12.136(C2×D4), C3⋊2(D4.5D4), C8.D6.2C2, (C2×Dic12)⋊21C2, C12.47D4⋊3C2, C6.49(C4⋊D4), C2.22(C12⋊D4), (C2×C24).102C22, (C2×C12).312C23, C4○D12.39C22, C22.6(Q8⋊3S3), (C2×Dic6).94C22, (C3×M4(2)).6C22, C4.Dic3.37C22, (C3×C8.C4)⋊6C2, (C2×C6).3(C4○D4), (C2×C4).113(C22×S3), SmallGroup(192,455)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.18D4
G = < a,b,c | a24=1, b4=c2=a12, bab-1=a7, cac-1=a-1, cbc-1=b3 >
Subgroups: 288 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, S3×C8, C8⋊S3, C24⋊C2, Dic12, C4.Dic3, C2×C24, C3×M4(2), C2×Dic6, C4○D12, D4.5D4, C12.47D4, C3×C8.C4, C8○D12, C2×Dic12, C8.D6, C24.18D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q8⋊3S3, D4.5D4, C12⋊D4, C24.18D4
Character table of C24.18D4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 12 | 24 | 24 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ24 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | -√2 | √2 | -√6 | √6 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | √2 | -√2 | -√6 | √6 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | √2 | -√2 | √6 | -√6 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | -√2 | √2 | √6 | -√6 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 96 points
Generators in S96
(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 53 81 45 13 65 93 33)(2 60 82 28 14 72 94 40)(3 67 83 35 15 55 95 47)(4 50 84 42 16 62 96 30)(5 57 85 25 17 69 73 37)(6 64 86 32 18 52 74 44)(7 71 87 39 19 59 75 27)(8 54 88 46 20 66 76 34)(9 61 89 29 21 49 77 41)(10 68 90 36 22 56 78 48)(11 51 91 43 23 63 79 31)(12 58 92 26 24 70 80 38)
(1 45 13 33)(2 44 14 32)(3 43 15 31)(4 42 16 30)(5 41 17 29)(6 40 18 28)(7 39 19 27)(8 38 20 26)(9 37 21 25)(10 36 22 48)(11 35 23 47)(12 34 24 46)(49 73 61 85)(50 96 62 84)(51 95 63 83)(52 94 64 82)(53 93 65 81)(54 92 66 80)(55 91 67 79)(56 90 68 78)(57 89 69 77)(58 88 70 76)(59 87 71 75)(60 86 72 74)
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), (1,53,81,45,13,65,93,33)(2,60,82,28,14,72,94,40)(3,67,83,35,15,55,95,47)(4,50,84,42,16,62,96,30)(5,57,85,25,17,69,73,37)(6,64,86,32,18,52,74,44)(7,71,87,39,19,59,75,27)(8,54,88,46,20,66,76,34)(9,61,89,29,21,49,77,41)(10,68,90,36,22,56,78,48)(11,51,91,43,23,63,79,31)(12,58,92,26,24,70,80,38), (1,45,13,33)(2,44,14,32)(3,43,15,31)(4,42,16,30)(5,41,17,29)(6,40,18,28)(7,39,19,27)(8,38,20,26)(9,37,21,25)(10,36,22,48)(11,35,23,47)(12,34,24,46)(49,73,61,85)(50,96,62,84)(51,95,63,83)(52,94,64,82)(53,93,65,81)(54,92,66,80)(55,91,67,79)(56,90,68,78)(57,89,69,77)(58,88,70,76)(59,87,71,75)(60,86,72,74)>;
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), (1,53,81,45,13,65,93,33)(2,60,82,28,14,72,94,40)(3,67,83,35,15,55,95,47)(4,50,84,42,16,62,96,30)(5,57,85,25,17,69,73,37)(6,64,86,32,18,52,74,44)(7,71,87,39,19,59,75,27)(8,54,88,46,20,66,76,34)(9,61,89,29,21,49,77,41)(10,68,90,36,22,56,78,48)(11,51,91,43,23,63,79,31)(12,58,92,26,24,70,80,38), (1,45,13,33)(2,44,14,32)(3,43,15,31)(4,42,16,30)(5,41,17,29)(6,40,18,28)(7,39,19,27)(8,38,20,26)(9,37,21,25)(10,36,22,48)(11,35,23,47)(12,34,24,46)(49,73,61,85)(50,96,62,84)(51,95,63,83)(52,94,64,82)(53,93,65,81)(54,92,66,80)(55,91,67,79)(56,90,68,78)(57,89,69,77)(58,88,70,76)(59,87,71,75)(60,86,72,74) );
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)], [(1,53,81,45,13,65,93,33),(2,60,82,28,14,72,94,40),(3,67,83,35,15,55,95,47),(4,50,84,42,16,62,96,30),(5,57,85,25,17,69,73,37),(6,64,86,32,18,52,74,44),(7,71,87,39,19,59,75,27),(8,54,88,46,20,66,76,34),(9,61,89,29,21,49,77,41),(10,68,90,36,22,56,78,48),(11,51,91,43,23,63,79,31),(12,58,92,26,24,70,80,38)], [(1,45,13,33),(2,44,14,32),(3,43,15,31),(4,42,16,30),(5,41,17,29),(6,40,18,28),(7,39,19,27),(8,38,20,26),(9,37,21,25),(10,36,22,48),(11,35,23,47),(12,34,24,46),(49,73,61,85),(50,96,62,84),(51,95,63,83),(52,94,64,82),(53,93,65,81),(54,92,66,80),(55,91,67,79),(56,90,68,78),(57,89,69,77),(58,88,70,76),(59,87,71,75),(60,86,72,74)]])
Matrix representation of C24.18D4 ►in GL4(𝔽73) generated by
| 0 | 16 | 0 | 57 |
| 57 | 16 | 16 | 57 |
| 0 | 16 | 0 | 16 |
| 57 | 16 | 57 | 16 |
| 14 | 12 | 65 | 14 |
| 61 | 26 | 59 | 6 |
| 65 | 14 | 59 | 61 |
| 59 | 6 | 12 | 47 |
| 12 | 14 | 14 | 65 |
| 26 | 61 | 6 | 59 |
| 14 | 65 | 61 | 59 |
| 6 | 59 | 47 | 12 |
G:=sub<GL(4,GF(73))| [0,57,0,57,16,16,16,16,0,16,0,57,57,57,16,16],[14,61,65,59,12,26,14,6,65,59,59,12,14,6,61,47],[12,26,14,6,14,61,65,59,14,6,61,47,65,59,59,12] >;
C24.18D4 in GAP, Magma, Sage, TeX
C_{24}._{18}D_4 % in TeX
G:=Group("C24.18D4"); // GroupNames label
G:=SmallGroup(192,455);
// by ID
G=gap.SmallGroup(192,455);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,58,1123,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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