direct product, abelian, monomial, 3-elementary
Aliases: C3×C24, SmallGroup(72,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C24 |
| C1 — C3×C24 |
| C1 — C3×C24 |
Generators and relations for C3×C24
G = < a,b | a3=b24=1, ab=ba >
Smallest permutation representation of C3×C24
►Regular action on 72 points
Generators in S72
(1 37 64)(2 38 65)(3 39 66)(4 40 67)(5 41 68)(6 42 69)(7 43 70)(8 44 71)(9 45 72)(10 46 49)(11 47 50)(12 48 51)(13 25 52)(14 26 53)(15 27 54)(16 28 55)(17 29 56)(18 30 57)(19 31 58)(20 32 59)(21 33 60)(22 34 61)(23 35 62)(24 36 63)
(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)
G:=sub<Sym(72)| (1,37,64)(2,38,65)(3,39,66)(4,40,67)(5,41,68)(6,42,69)(7,43,70)(8,44,71)(9,45,72)(10,46,49)(11,47,50)(12,48,51)(13,25,52)(14,26,53)(15,27,54)(16,28,55)(17,29,56)(18,30,57)(19,31,58)(20,32,59)(21,33,60)(22,34,61)(23,35,62)(24,36,63), (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)>;
G:=Group( (1,37,64)(2,38,65)(3,39,66)(4,40,67)(5,41,68)(6,42,69)(7,43,70)(8,44,71)(9,45,72)(10,46,49)(11,47,50)(12,48,51)(13,25,52)(14,26,53)(15,27,54)(16,28,55)(17,29,56)(18,30,57)(19,31,58)(20,32,59)(21,33,60)(22,34,61)(23,35,62)(24,36,63), (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) );
G=PermutationGroup([[(1,37,64),(2,38,65),(3,39,66),(4,40,67),(5,41,68),(6,42,69),(7,43,70),(8,44,71),(9,45,72),(10,46,49),(11,47,50),(12,48,51),(13,25,52),(14,26,53),(15,27,54),(16,28,55),(17,29,56),(18,30,57),(19,31,58),(20,32,59),(21,33,60),(22,34,61),(23,35,62),(24,36,63)], [(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)]])
C3×C24 is a maximal subgroup of
C24.S3 C24⋊S3 C24⋊2S3 C32⋊5D8 C32⋊5Q16
72 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 |
| kernel | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C3×C24 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 1 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 63 |
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,1],[9,0,0,0,9,0,0,0,63] >;
C3×C24 in GAP, Magma, Sage, TeX
C_3\times C_{24} % in TeX
G:=Group("C3xC24"); // GroupNames label
G:=SmallGroup(72,14);
// by ID
G=gap.SmallGroup(72,14);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-2,90,58]);
// Polycyclic
G:=Group<a,b|a^3=b^24=1,a*b=b*a>;
// generators/relations
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