C₃×D₃×C₃⋊AGL(1,ℤ/9) #72243297

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃²

Aliases

Fitting, Sylow	C₃³×3₋¹⁺² ⋊ C₂²
derived	C₃²×C₉ ⋊ C₂²×C₃²

	subgroup	G/H
center, hypercenter	C₃	D₃×C₃⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenter	C₃²×C₉	C₂²×C₃²
Frattini	C₃	C₃²×D₃×C₃⋊D₃
Fitting	C₃³×3₋¹⁺²	C₂²
socle	C₃⁴	C₂×C₃×D₃
LCS	G ▷ C₃²×C₉
UCS	C₁ ◁ C₃

2068 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 2068 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

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

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 18 points:

(5,6)(7,8,9)(11,17,18,13,14,15)(12,16);(1,2,3)(4,5,6)(8,9);(10,14,13,12,18,15,16,11,17)

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((5,6)(7,8,9)(11,17,18,13,14,15)(12,16), (1,2,3)(4,5,6)(8,9), (10,14,13,12,18,15,16,11,17));