C₅×#554 #66218

order	320
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅
prime radical	C₂×C₅×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1556);