Copied to
clipboard

G = C27⋊C3order 81 = 34

The semidirect product of C27 and C3 acting faithfully

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: C27⋊C3, C9.C9, C32.C9, C9.2C32, C3.3(C3×C9), (C3×C9).3C3, SmallGroup(81,6)

Series: Derived Chief Lower central Upper central Jennings

C1C3 — C27⋊C3
C1C3C9C3×C9 — C27⋊C3
C1C3 — C27⋊C3
C1C9 — C27⋊C3
C1C3C3C3C3C3C3C9C9 — C27⋊C3

Generators and relations for C27⋊C3
 G = < a,b | a27=b3=1, bab-1=a10 >

3C3

Permutation representations of C27⋊C3
On 27 points - transitive group 27T22
Generators in S27
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)

G:=sub<Sym(27)| (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)>;

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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27) );

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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27)])

G:=TransitiveGroup(27,22);

C27⋊C3 is a maximal subgroup of
C27⋊C6  C9.4He3  C9.5He3  C9.6He3  C27⋊C9  C27○He3  C27⋊A4  C62.C9
C27⋊C3 is a maximal quotient of
C272C9  C32⋊C27  C9⋊C27  C27⋊A4  C62.C9

33 conjugacy classes

class 1 3A3B3C3D9A···9F9G9H9I9J27A···27R
order133339···9999927···27
size111331···133333···3

33 irreducible representations

dim111113
type+
imageC1C3C3C9C9C27⋊C3
kernelC27⋊C3C27C3×C9C9C32C1
# reps1621266

Matrix representation of C27⋊C3 in GL3(𝔽109) generated by

010
0063
1600
,
100
0630
0045
G:=sub<GL(3,GF(109))| [0,0,16,1,0,0,0,63,0],[1,0,0,0,63,0,0,0,45] >;

C27⋊C3 in GAP, Magma, Sage, TeX

C_{27}\rtimes C_3
% in TeX

G:=Group("C27:C3");
// GroupNames label

G:=SmallGroup(81,6);
// by ID

G=gap.SmallGroup(81,6);
# by ID

G:=PCGroup([4,-3,3,-3,-3,36,241,46]);
// Polycyclic

G:=Group<a,b|a^27=b^3=1,b*a*b^-1=a^10>;
// generators/relations

Export

Subgroup lattice of C27⋊C3 in TeX

׿
×
𝔽