D₅×((C₂²⋊C₉)⋊C₂)⋊C₃ #72247637

order	2160
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

derived	C₅×C₂²⋊C₉ ⋊ C₂²×C₃
Sylow	C₅ ⋊ C₂×((C₂²⋊C₉)⋊C₂)⋊C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×C₂²⋊C₉	C₂²×C₃
Frattini	C₃	C₃×D₅×S₄
Fitting	C₂²×C₃²×C₅	C₂×D₃
socle	C₂²×C₃×C₅	C₂×C₃×D₃
LCS	G ▷ C₅×C₂²⋊C₉
UCS	C₁

240 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 90

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

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

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

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