A₄×D₉×AGL(1,𝔽₅) #72247688

order	4320
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Fitting, derived	C₂²×C₅×C₉ ⋊ C₂×C₃×C₄
Sylow	C₅ ⋊ C₄×A₄×D₉

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₂²×C₅×C₉	C₂×C₃×C₄
Frattini	C₃	D₃×A₄×AGL(1,𝔽₅)
socle	C₂²×C₃×C₅	C₃×C₄×D₃
prime radical	D₅×A₄×D₉	C₂
cosocle	C₂²×C₉×D₅	C₂²×C₃
LCS	G ▷ C₂²×C₅×C₉
UCS	C₁

420 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 54 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)(6,13,14,8,7,9,12,11,10)(16,18,17);(1,2,5,3,4)(7,9)(8,12)(10,13)(11,14)(15,18)(16,17)

maximal subgroup of	A₄×AGL(1,𝔽₅)×AGL(1,ℤ/9), D₉×AGL(1,𝔽₅)×S₄
maximal normal subgroup of	A₄×AGL(1,𝔽₅)×AGL(1,ℤ/9), D₉×AGL(1,𝔽₅)×S₄

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

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