A₄×C₅⋊AGL(1,ℤ/9) #72247486

order	3240
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×C₅×C₉	C₂×C₃²
Frattini	C₃	C₃×A₄×D₁₅
Fitting	C₂²×C₅×3₋¹⁺²	C₂×C₃
socle	C₂²×C₃×C₅	C₃²×D₃
LCS	G ▷ C₂²×C₅×C₉
UCS	C₁

234 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

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

maximal subgroup of	D₅×A₄×AGL(1,ℤ/9), S₄×C₅⋊AGL(1,ℤ/9), #72254734
maximal normal subgroup of	D₅×A₄×AGL(1,ℤ/9), S₄×C₅⋊AGL(1,ℤ/9), #72254734

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

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