metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.99D4, M4(2).3Dic3, D4.(C3⋊C8), C12.9(C2×C8), C3⋊3(D4.C8), C8○D4.4S3, (C3×D4).1C8, Q8.2(C3⋊C8), (C3×Q8).1C8, (C2×C8).268D6, C8.34(C3⋊D4), C4○D4.4Dic3, C12.C8⋊12C2, (C2×C6).6M4(2), C6.19(C22⋊C8), (C3×M4(2)).5C4, C12.96(C22⋊C4), (C2×C24).266C22, C2.8(C12.55D4), C4.30(C6.D4), C22.1(C4.Dic3), C4.3(C2×C3⋊C8), (C2×C3⋊C16)⋊14C2, (C3×C4○D4).1C4, (C3×C8○D4).3C2, (C2×C12).67(C2×C4), (C2×C4).41(C2×Dic3), SmallGroup(192,120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.99D4
G = < a,b,c | a24=1, b4=a6, c2=a9, bab-1=cac-1=a17, cbc-1=a3b3 >
Smallest permutation representation of C24.99D4
►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 75 66 39 7 81 72 45 13 87 54 27 19 93 60 33)(2 92 67 32 8 74 49 38 14 80 55 44 20 86 61 26)(3 85 68 25 9 91 50 31 15 73 56 37 21 79 62 43)(4 78 69 42 10 84 51 48 16 90 57 30 22 96 63 36)(5 95 70 35 11 77 52 41 17 83 58 47 23 89 64 29)(6 88 71 28 12 94 53 34 18 76 59 40 24 82 65 46)
(1 42 10 27 19 36 4 45 13 30 22 39 7 48 16 33)(2 35 11 44 20 29 5 38 14 47 23 32 8 41 17 26)(3 28 12 37 21 46 6 31 15 40 24 25 9 34 18 43)(49 95 58 80 67 89 52 74 61 83 70 92 55 77 64 86)(50 88 59 73 68 82 53 91 62 76 71 85 56 94 65 79)(51 81 60 90 69 75 54 84 63 93 72 78 57 87 66 96)
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), (1,75,66,39,7,81,72,45,13,87,54,27,19,93,60,33)(2,92,67,32,8,74,49,38,14,80,55,44,20,86,61,26)(3,85,68,25,9,91,50,31,15,73,56,37,21,79,62,43)(4,78,69,42,10,84,51,48,16,90,57,30,22,96,63,36)(5,95,70,35,11,77,52,41,17,83,58,47,23,89,64,29)(6,88,71,28,12,94,53,34,18,76,59,40,24,82,65,46), (1,42,10,27,19,36,4,45,13,30,22,39,7,48,16,33)(2,35,11,44,20,29,5,38,14,47,23,32,8,41,17,26)(3,28,12,37,21,46,6,31,15,40,24,25,9,34,18,43)(49,95,58,80,67,89,52,74,61,83,70,92,55,77,64,86)(50,88,59,73,68,82,53,91,62,76,71,85,56,94,65,79)(51,81,60,90,69,75,54,84,63,93,72,78,57,87,66,96)>;
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), (1,75,66,39,7,81,72,45,13,87,54,27,19,93,60,33)(2,92,67,32,8,74,49,38,14,80,55,44,20,86,61,26)(3,85,68,25,9,91,50,31,15,73,56,37,21,79,62,43)(4,78,69,42,10,84,51,48,16,90,57,30,22,96,63,36)(5,95,70,35,11,77,52,41,17,83,58,47,23,89,64,29)(6,88,71,28,12,94,53,34,18,76,59,40,24,82,65,46), (1,42,10,27,19,36,4,45,13,30,22,39,7,48,16,33)(2,35,11,44,20,29,5,38,14,47,23,32,8,41,17,26)(3,28,12,37,21,46,6,31,15,40,24,25,9,34,18,43)(49,95,58,80,67,89,52,74,61,83,70,92,55,77,64,86)(50,88,59,73,68,82,53,91,62,76,71,85,56,94,65,79)(51,81,60,90,69,75,54,84,63,93,72,78,57,87,66,96) );
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)], [(1,75,66,39,7,81,72,45,13,87,54,27,19,93,60,33),(2,92,67,32,8,74,49,38,14,80,55,44,20,86,61,26),(3,85,68,25,9,91,50,31,15,73,56,37,21,79,62,43),(4,78,69,42,10,84,51,48,16,90,57,30,22,96,63,36),(5,95,70,35,11,77,52,41,17,83,58,47,23,89,64,29),(6,88,71,28,12,94,53,34,18,76,59,40,24,82,65,46)], [(1,42,10,27,19,36,4,45,13,30,22,39,7,48,16,33),(2,35,11,44,20,29,5,38,14,47,23,32,8,41,17,26),(3,28,12,37,21,46,6,31,15,40,24,25,9,34,18,43),(49,95,58,80,67,89,52,74,61,83,70,92,55,77,64,86),(50,88,59,73,68,82,53,91,62,76,71,85,56,94,65,79),(51,81,60,90,69,75,54,84,63,93,72,78,57,87,66,96)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D4 | D6 | Dic3 | Dic3 | M4(2) | C3⋊D4 | C3⋊C8 | C3⋊C8 | C4.Dic3 | D4.C8 | C24.99D4 |
| kernel | C24.99D4 | C2×C3⋊C16 | C12.C8 | C3×C8○D4 | C3×M4(2) | C3×C4○D4 | C3×D4 | C3×Q8 | C8○D4 | C24 | C2×C8 | M4(2) | C4○D4 | C2×C6 | C8 | D4 | Q8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C24.99D4 ►in GL4(𝔽97) generated by
| 33 | 0 | 0 | 0 |
| 0 | 33 | 0 | 0 |
| 0 | 0 | 35 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 39 | 0 | 0 |
| 29 | 39 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 96 | 0 |
| 0 | 39 | 0 | 0 |
| 68 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(97))| [33,0,0,0,0,33,0,0,0,0,35,0,0,0,0,61],[0,29,0,0,39,39,0,0,0,0,0,96,0,0,1,0],[0,68,0,0,39,0,0,0,0,0,0,1,0,0,1,0] >;
C24.99D4 in GAP, Magma, Sage, TeX
C_{24}._{99}D_4 % in TeX
G:=Group("C24.99D4"); // GroupNames label
G:=SmallGroup(192,120);
// by ID
G=gap.SmallGroup(192,120);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,100,1123,570,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^3*b^3>;
// generators/relations
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