direct product, cyclic, abelian, monomial
Aliases: C75, also denoted Z75, SmallGroup(75,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C75 |
| C1 — C75 |
| C1 — C75 |
Generators and relations for C75
G = < a | a75=1 >
Smallest permutation representation of C75
►Regular action on 75 points
Generators in S75
(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)
G:=sub<Sym(75)| (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)>;
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) );
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)]])
C75 is a maximal subgroup of
D75
75 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 15A | ··· | 15H | 25A | ··· | 25T | 75A | ··· | 75AN |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 | 25 | ··· | 25 | 75 | ··· | 75 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C25 | C75 |
| kernel | C75 | C25 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 8 | 20 | 40 |
Matrix representation of C75 ►in GL1(𝔽151) generated by
| 47 |
G:=sub<GL(1,GF(151))| [47] >;
C75 in GAP, Magma, Sage, TeX
C_{75} % in TeX
G:=Group("C75"); // GroupNames label
G:=SmallGroup(75,1);
// by ID
G=gap.SmallGroup(75,1);
# by ID
G:=PCGroup([3,-3,-5,-5,34]);
// Polycyclic
G:=Group<a|a^75=1>;
// generators/relations
Export