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G = C84order 84 = 22·3·7

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C84, also denoted Z84, SmallGroup(84,6)

Series: Derived Chief Lower central Upper central

C1 — C84
C1C2C14C42 — 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 3A3B4A4B6A6B7A···7F12A12B12C12D14A···14F21A···21L28A···28L42A···42L84A···84X
order123344667···71212121214···1421···2128···2842···4284···84
size111111111···111111···11···11···11···11···1

84 irreducible representations

dim111111111111
type++
imageC1C2C3C4C6C7C12C14C21C28C42C84
kernelC84C42C28C21C14C12C7C6C4C3C2C1
# reps1122264612121224

Matrix representation of C84 in GL2(𝔽13) generated by

09
16
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

Export

Subgroup lattice of C84 in TeX

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