order	320
isoclinism	isoclinic class of (C₂⁴⋊C₅)⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×(C₂⁴⋊C₅)⋊C₂
unambiguous semidirect product	(C₂⁴⋊C₅)⋊C₂²
unambiguous semidirect product	(C₂×C₂⁴⋊C₅)⋊C₂

Splittings

Fitting, socle	C₂⁵ ⋊ D₅
derived	C₂⁴⋊C₅ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	(C₂⁴⋊C₅)⋊C₂
commutator, cosocle, hypocenter	C₂⁴⋊C₅	C₂²
Frattini	C₁	G
Fitting, socle	C₂⁵	D₅
LCS	G ▷ C₂⁴⋊C₅
UCS	C₁ ◁ C₂

193 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2)(3,5,4,7)(6,10,8,9);(1,4,6,9,7,2,3,8,10,5)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1636);