C₉×#6899 #14552042

order	2187
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃³×C₉ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₉×#6899

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

947 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 947 conjugacy classes of subgroups — too many to draw. Only its 494 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) = 45
transitive degree	μ_t(G) = 729

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5,6,7,8,9);(11,14,12)(15,16,17)(19,20,24)(21,28,40)(22,26,34)(23,27,35)(25,36,31)(29,39,43)(30,41,33)(32,38,44)(37,45,42);(10,11,15,18,12,16,13,14,17)(19,21,29,23,30,42,32,31,22)(20,25,37,27,28,26,38,41,39)(24,33,34,35,36,43,44,40,45);(10,12,17,18,14,15,13,11,16)(19,22,31,32,42,30,23,29,21)(20,26,25,38,37,41,27,39,28)(24,34,36,44,45,33,35,43,40)

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, 5901);