metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.98D4, C6.6M5(2), C12.29M4(2), (C2xC6):2C16, (C2xC12).5C8, C22:2(C3:C16), C3:2(C22:C16), C6.10(C2xC16), (C2xC24).11C4, (C2xC8).336D6, C23.3(C3:C8), (C22xC6).5C8, (C22xC8).3S3, (C2xC8).6Dic3, C8.52(C3:D4), C6.17(C22:C8), (C22xC24).25C2, (C22xC12).29C4, C2.3(C12.C8), C12.92(C22:C4), (C2xC24).422C22, C4.11(C4.Dic3), (C22xC4).11Dic3, C4.24(C6.D4), C2.2(C12.55D4), C2.5(C2xC3:C16), (C2xC3:C16):10C2, (C2xC4).4(C3:C8), (C2xC6).31(C2xC8), C22.11(C2xC3:C8), (C2xC12).312(C2xC4), (C2xC4).93(C2xDic3), SmallGroup(192,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.98D4
G = < a,b,c | a24=1, b4=a18, c2=a9, bab-1=cac-1=a17, cbc-1=a15b3 >
Subgroups: 104 in 66 conjugacy classes, 39 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, C23, C12, C12, C2xC6, C2xC6, C2xC6, C16, C2xC8, C2xC8, C22xC4, C24, C24, C2xC12, C2xC12, C22xC6, C2xC16, C22xC8, C3:C16, C2xC24, C2xC24, C22xC12, C22:C16, C2xC3:C16, C22xC24, C24.98D4
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, Dic3, D6, C16, C22:C4, C2xC8, M4(2), C3:C8, C2xDic3, C3:D4, C22:C8, C2xC16, M5(2), C3:C16, C2xC3:C8, C4.Dic3, C6.D4, C22:C16, C2xC3:C16, C12.C8, C12.55D4, C24.98D4
►On 96 points
(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 35 50 91 19 29 68 85 13 47 62 79 7 41 56 73)(2 28 51 84 20 46 69 78 14 40 63 96 8 34 57 90)(3 45 52 77 21 39 70 95 15 33 64 89 9 27 58 83)(4 38 53 94 22 32 71 88 16 26 65 82 10 44 59 76)(5 31 54 87 23 25 72 81 17 43 66 75 11 37 60 93)(6 48 55 80 24 42 49 74 18 36 67 92 12 30 61 86)
(1 82 10 91 19 76 4 85 13 94 22 79 7 88 16 73)(2 75 11 84 20 93 5 78 14 87 23 96 8 81 17 90)(3 92 12 77 21 86 6 95 15 80 24 89 9 74 18 83)(25 72 34 57 43 66 28 51 37 60 46 69 31 54 40 63)(26 65 35 50 44 59 29 68 38 53 47 62 32 71 41 56)(27 58 36 67 45 52 30 61 39 70 48 55 33 64 42 49)
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), (1,35,50,91,19,29,68,85,13,47,62,79,7,41,56,73)(2,28,51,84,20,46,69,78,14,40,63,96,8,34,57,90)(3,45,52,77,21,39,70,95,15,33,64,89,9,27,58,83)(4,38,53,94,22,32,71,88,16,26,65,82,10,44,59,76)(5,31,54,87,23,25,72,81,17,43,66,75,11,37,60,93)(6,48,55,80,24,42,49,74,18,36,67,92,12,30,61,86), (1,82,10,91,19,76,4,85,13,94,22,79,7,88,16,73)(2,75,11,84,20,93,5,78,14,87,23,96,8,81,17,90)(3,92,12,77,21,86,6,95,15,80,24,89,9,74,18,83)(25,72,34,57,43,66,28,51,37,60,46,69,31,54,40,63)(26,65,35,50,44,59,29,68,38,53,47,62,32,71,41,56)(27,58,36,67,45,52,30,61,39,70,48,55,33,64,42,49)>;
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), (1,35,50,91,19,29,68,85,13,47,62,79,7,41,56,73)(2,28,51,84,20,46,69,78,14,40,63,96,8,34,57,90)(3,45,52,77,21,39,70,95,15,33,64,89,9,27,58,83)(4,38,53,94,22,32,71,88,16,26,65,82,10,44,59,76)(5,31,54,87,23,25,72,81,17,43,66,75,11,37,60,93)(6,48,55,80,24,42,49,74,18,36,67,92,12,30,61,86), (1,82,10,91,19,76,4,85,13,94,22,79,7,88,16,73)(2,75,11,84,20,93,5,78,14,87,23,96,8,81,17,90)(3,92,12,77,21,86,6,95,15,80,24,89,9,74,18,83)(25,72,34,57,43,66,28,51,37,60,46,69,31,54,40,63)(26,65,35,50,44,59,29,68,38,53,47,62,32,71,41,56)(27,58,36,67,45,52,30,61,39,70,48,55,33,64,42,49) );
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)], [(1,35,50,91,19,29,68,85,13,47,62,79,7,41,56,73),(2,28,51,84,20,46,69,78,14,40,63,96,8,34,57,90),(3,45,52,77,21,39,70,95,15,33,64,89,9,27,58,83),(4,38,53,94,22,32,71,88,16,26,65,82,10,44,59,76),(5,31,54,87,23,25,72,81,17,43,66,75,11,37,60,93),(6,48,55,80,24,42,49,74,18,36,67,92,12,30,61,86)], [(1,82,10,91,19,76,4,85,13,94,22,79,7,88,16,73),(2,75,11,84,20,93,5,78,14,87,23,96,8,81,17,90),(3,92,12,77,21,86,6,95,15,80,24,89,9,74,18,83),(25,72,34,57,43,66,28,51,37,60,46,69,31,54,40,63),(26,65,35,50,44,59,29,68,38,53,47,62,32,71,41,56),(27,58,36,67,45,52,30,61,39,70,48,55,33,64,42,49)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 16A | ··· | 16P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D4 | Dic3 | D6 | Dic3 | M4(2) | C3:D4 | C3:C8 | C3:C8 | M5(2) | C4.Dic3 | C3:C16 | C12.C8 |
| kernel | C24.98D4 | C2xC3:C16 | C22xC24 | C2xC24 | C22xC12 | C2xC12 | C22xC6 | C2xC6 | C22xC8 | C24 | C2xC8 | C2xC8 | C22xC4 | C12 | C8 | C2xC4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C24.98D4 ►in GL3(F97) generated by
| 64 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 43 |
| 8 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 47 | 0 |
| 89 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 50 | 0 |
G:=sub<GL(3,GF(97))| [64,0,0,0,4,0,0,0,43],[8,0,0,0,0,47,0,1,0],[89,0,0,0,0,50,0,1,0] >;
C24.98D4 in GAP, Magma, Sage, TeX
C_{24}._{98}D_4 % in TeX
G:=Group("C24.98D4"); // GroupNames label
G:=SmallGroup(192,108);
// by ID
G=gap.SmallGroup(192,108);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,100,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=a^18,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^15*b^3>;
// generators/relations