direct product, cyclic, abelian, monomial
Aliases: C70, also denoted Z70, SmallGroup(70,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C70 |
| C1 — C70 |
| C1 — C70 |
Generators and relations for C70
G = < a | a70=1 >
Smallest permutation representation of C70
►Regular action on 70 points
Generators in S70
(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)
G:=sub<Sym(70)| (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)>;
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) );
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)]])
C70 is a maximal subgroup of
Dic35
70 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | 10B | 10C | 10D | 14A | ··· | 14F | 35A | ··· | 35X | 70A | ··· | 70X |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 |
| kernel | C70 | C35 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 4 | 6 | 4 | 6 | 24 | 24 |
Matrix representation of C70 ►in GL1(𝔽71) generated by
| 62 |
G:=sub<GL(1,GF(71))| [62] >;
C70 in GAP, Magma, Sage, TeX
C_{70} % in TeX
G:=Group("C70"); // GroupNames label
G:=SmallGroup(70,4);
// by ID
G=gap.SmallGroup(70,4);
# by ID
G:=PCGroup([3,-2,-5,-7]);
// Polycyclic
G:=Group<a|a^70=1>;
// generators/relations
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