direct product, abelian, monomial, 2-elementary
Aliases: C4xC24, SmallGroup(96,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4xC24 |
| C1 — C4xC24 |
| C1 — C4xC24 |
Generators and relations for C4xC24
G = < a,b | a4=b24=1, ab=ba >
Quotients: C1, C2, C3, C4, C22, C6, C8, C2xC4, C12, C2xC6, C42, C2xC8, C24, C2xC12, C4xC8, C4xC12, C2xC24, C4xC24
Smallest permutation representation of C4xC24
►Regular action on 96 points
Generators in S96
(1 41 68 80)(2 42 69 81)(3 43 70 82)(4 44 71 83)(5 45 72 84)(6 46 49 85)(7 47 50 86)(8 48 51 87)(9 25 52 88)(10 26 53 89)(11 27 54 90)(12 28 55 91)(13 29 56 92)(14 30 57 93)(15 31 58 94)(16 32 59 95)(17 33 60 96)(18 34 61 73)(19 35 62 74)(20 36 63 75)(21 37 64 76)(22 38 65 77)(23 39 66 78)(24 40 67 79)
(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)
G:=sub<Sym(96)| (1,41,68,80)(2,42,69,81)(3,43,70,82)(4,44,71,83)(5,45,72,84)(6,46,49,85)(7,47,50,86)(8,48,51,87)(9,25,52,88)(10,26,53,89)(11,27,54,90)(12,28,55,91)(13,29,56,92)(14,30,57,93)(15,31,58,94)(16,32,59,95)(17,33,60,96)(18,34,61,73)(19,35,62,74)(20,36,63,75)(21,37,64,76)(22,38,65,77)(23,39,66,78)(24,40,67,79), (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)>;
G:=Group( (1,41,68,80)(2,42,69,81)(3,43,70,82)(4,44,71,83)(5,45,72,84)(6,46,49,85)(7,47,50,86)(8,48,51,87)(9,25,52,88)(10,26,53,89)(11,27,54,90)(12,28,55,91)(13,29,56,92)(14,30,57,93)(15,31,58,94)(16,32,59,95)(17,33,60,96)(18,34,61,73)(19,35,62,74)(20,36,63,75)(21,37,64,76)(22,38,65,77)(23,39,66,78)(24,40,67,79), (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) );
G=PermutationGroup([[(1,41,68,80),(2,42,69,81),(3,43,70,82),(4,44,71,83),(5,45,72,84),(6,46,49,85),(7,47,50,86),(8,48,51,87),(9,25,52,88),(10,26,53,89),(11,27,54,90),(12,28,55,91),(13,29,56,92),(14,30,57,93),(15,31,58,94),(16,32,59,95),(17,33,60,96),(18,34,61,73),(19,35,62,74),(20,36,63,75),(21,37,64,76),(22,38,65,77),(23,39,66,78),(24,40,67,79)], [(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)]])
C4xC24 is a maximal subgroup of
C42.279D6 C24:C8 C4.8Dic12 C24:2C8 C24:1C8 C4.17D24 C24.C8 C12:C16 C24.1C8 C24:12Q8 C24:9Q8 C12.14Q16 C24:8Q8 C24.13Q8 C42.282D6 C8:6D12 D6.C42 C42.243D6 C8:5D12 C4.5D24 C12:4D8 C8.8D12 C42.264D6 C12:4Q16 D24:11C4
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 8A | ··· | 8P | 12A | ··· | 12X | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 |
| kernel | C4xC24 | C4xC12 | C2xC24 | C4xC8 | C24 | C2xC12 | C42 | C2xC8 | C12 | C8 | C2xC4 | C4 |
| # reps | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 16 | 16 | 8 | 32 |
Matrix representation of C4xC24 ►in GL2(F73) generated by
| 46 | 0 |
| 0 | 72 |
| 51 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(73))| [46,0,0,72],[51,0,0,21] >;
C4xC24 in GAP, Magma, Sage, TeX
C_4\times C_{24} % in TeX
G:=Group("C4xC24"); // GroupNames label
G:=SmallGroup(96,46);
// by ID
G=gap.SmallGroup(96,46);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,151,117]);
// Polycyclic
G:=Group<a,b|a^4=b^24=1,a*b=b*a>;
// generators/relations
Export