C₅×#1613 #10603121

order	640
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
prime radical	C₅×#534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3,4,5)(6,7,9,12)(8,10,13,11)(15,18)(16,20)(19,21);(7,10)(8,13)(11,12)(14,15)(16,21)(17,19)(18,20);(16,20)(18,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5203);