C₂²×C₁₁⋊C₅ #6353

order	220
isoclinism	isoclinic class of C₁₁⋊C₅
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₁) (order 660) · Inn(G) = C₁₁⋊C₅ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×C₁₁⋊C₅
unambiguous semidirect product	(C₂²×C₁₁)⋊C₅

Fitting, socle, Hall	C₂²×C₁₁ ⋊ C₅
derived, Sylow	C₁₁ ⋊ C₂²×C₅

	subgroup	G/H
center, hypercenter	C₂²	C₁₁⋊C₅
commutator, cosocle, hypocenter	C₁₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁	C₅
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂²

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 44

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 8);