Copied to
clipboard

G = C92order 81 = 34

Abelian group of type [9,9]

direct product, p-group, abelian, monomial

Aliases: C92, SmallGroup(81,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C92
C1C3C32C3×C9 — C92
C1 — C92
C1 — C92
C1C32C32 — C92

Generators and relations for C92
 G = < a,b | a9=b9=1, ab=ba >


Smallest permutation representation of C92
Regular action on 81 points
Generators in S81
(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)
(1 60 33 80 70 53 26 14 40)(2 61 34 81 71 54 27 15 41)(3 62 35 73 72 46 19 16 42)(4 63 36 74 64 47 20 17 43)(5 55 28 75 65 48 21 18 44)(6 56 29 76 66 49 22 10 45)(7 57 30 77 67 50 23 11 37)(8 58 31 78 68 51 24 12 38)(9 59 32 79 69 52 25 13 39)

G:=sub<Sym(81)| (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), (1,60,33,80,70,53,26,14,40)(2,61,34,81,71,54,27,15,41)(3,62,35,73,72,46,19,16,42)(4,63,36,74,64,47,20,17,43)(5,55,28,75,65,48,21,18,44)(6,56,29,76,66,49,22,10,45)(7,57,30,77,67,50,23,11,37)(8,58,31,78,68,51,24,12,38)(9,59,32,79,69,52,25,13,39)>;

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), (1,60,33,80,70,53,26,14,40)(2,61,34,81,71,54,27,15,41)(3,62,35,73,72,46,19,16,42)(4,63,36,74,64,47,20,17,43)(5,55,28,75,65,48,21,18,44)(6,56,29,76,66,49,22,10,45)(7,57,30,77,67,50,23,11,37)(8,58,31,78,68,51,24,12,38)(9,59,32,79,69,52,25,13,39) );

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)], [(1,60,33,80,70,53,26,14,40),(2,61,34,81,71,54,27,15,41),(3,62,35,73,72,46,19,16,42),(4,63,36,74,64,47,20,17,43),(5,55,28,75,65,48,21,18,44),(6,56,29,76,66,49,22,10,45),(7,57,30,77,67,50,23,11,37),(8,58,31,78,68,51,24,12,38),(9,59,32,79,69,52,25,13,39)])

C92 is a maximal subgroup of
C9⋊D9  C272C9  C9⋊C27  C92⋊C3  C922C3  C92.C3  C923C3  C927C3  C924C3  C925C3  C928C3  C929C3
C92 is a maximal quotient of
C3.C92  C272C9

81 conjugacy classes

class 1 3A···3H9A···9BT
order13···39···9
size11···11···1

81 irreducible representations

dim111
type+
imageC1C3C9
kernelC92C3×C9C9
# reps1872

Matrix representation of C92 in GL2(𝔽19) generated by

110
016
,
60
01
G:=sub<GL(2,GF(19))| [11,0,0,16],[6,0,0,1] >;

C92 in GAP, Magma, Sage, TeX

C_9^2
% in TeX

G:=Group("C9^2");
// GroupNames label

G:=SmallGroup(81,2);
// by ID

G=gap.SmallGroup(81,2);
# by ID

G:=PCGroup([4,-3,3,-3,3,36,77]);
// Polycyclic

G:=Group<a,b|a^9=b^9=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C92 in TeX

׿
×
𝔽