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G = C72order 72 = 23·32

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C72, also denoted Z72, SmallGroup(72,2)

Series: Derived Chief Lower central Upper central

C1 — C72
C1C2C6C12C36 — C72
C1 — C72
C1 — C72

Generators and relations for C72
 G = < a | a72=1 >


Smallest permutation representation of C72
Regular action on 72 points
Generators in S72
(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)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)])

C72 is a maximal subgroup of   C9⋊C16  Dic36  C8⋊D9  C72⋊C2  D72  Q8.C36

72 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D9A···9F12A12B12C12D18A···18F24A···24H36A···36L72A···72X
order1233446688889···91212121218···1824···2436···3672···72
size1111111111111···111111···11···11···11···1

72 irreducible representations

dim111111111111
type++
imageC1C2C3C4C6C8C9C12C18C24C36C72
kernelC72C36C24C18C12C9C8C6C4C3C2C1
# reps11222464681224

Matrix representation of C72 in GL1(𝔽73) generated by

28
G:=sub<GL(1,GF(73))| [28] >;

C72 in GAP, Magma, Sage, TeX

C_{72}
% in TeX

G:=Group("C72");
// GroupNames label

G:=SmallGroup(72,2);
// by ID

G=gap.SmallGroup(72,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-2,30,66,102]);
// Polycyclic

G:=Group<a|a^72=1>;
// generators/relations

Export

Subgroup lattice of C72 in TeX

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