C₉×#6896 #14552041

order	2187
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃³×C₉ · Schur multiplier (H₂) = C₃³×C₉

Aliases

direct product

C₉×#6896

Subgroups

	subgroup	G/H
center	C₃²×C₉	C₃³
commutator	C₃²	C₃³×C₉
Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₁

1028 conjugacy classes of subgroups; only the 521 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 521 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 729

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2,3,4,5,6,7,8,9);(11,15,17)(12,16,18)(20,21,25)(23,26,24);(10,11,16,14,17,18,13,15,12)(19,20,23,22,21,24,27,25,26);(10,12,15,13,18,17,14,16,11)(19,21,26,22,25,23,27,20,24)

Conjugacy classes & character table

459 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 5900);