A₄×#87831 #72231365

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Hall	C₂²×#87831 ⋊ C₃
Sylow	C₅² ⋊ A₄×C₂²⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄×D₅⋊AGL(1,𝔽₅)
commutator, Fitting, socle	C₂³×C₅²	C₂×C₃×C₄
prime radical	C₂×D₅²×A₄	C₂
cosocle	C₂³×C₅⋊D₅	C₂²×C₃
hypocenter	C₂²×C₅²	C₃×C₂²⋊C₄
LCS	G ▷ C₂³×C₅² ▷ C₂²×C₅²
UCS	C₁ ◁ C₂

808 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 60

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

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

maximal subgroup of	A₄×#11717375
maximal normal subgroup of	A₄×#11717375

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

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