direct product, cyclic, abelian, monomial
Aliases: C84, also denoted Z84, SmallGroup(84,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C84 |
| C1 — C84 |
| C1 — C84 |
Generators and relations for C84
G = < a | a84=1 >
Smallest permutation representation of C84
►Regular action on 84 points
Generators in S84
(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)
G:=sub<Sym(84)| (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)>;
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) );
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)]])
C84 is a maximal subgroup of
C21⋊C8 Dic42 D84
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 21A | ··· | 21L | 28A | ··· | 28L | 42A | ··· | 42L | 84A | ··· | 84X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C7 | C12 | C14 | C21 | C28 | C42 | C84 |
| kernel | C84 | C42 | C28 | C21 | C14 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C84 ►in GL2(𝔽13) generated by
| 0 | 9 |
| 1 | 6 |
G:=sub<GL(2,GF(13))| [0,1,9,6] >;
C84 in GAP, Magma, Sage, TeX
C_{84} % in TeX
G:=Group("C84"); // GroupNames label
G:=SmallGroup(84,6);
// by ID
G=gap.SmallGroup(84,6);
# by ID
G:=PCGroup([4,-2,-3,-7,-2,168]);
// Polycyclic
G:=Group<a|a^84=1>;
// generators/relations
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