(C₁₁⋊C₅)⋊C₈ #89148

order	440
isoclinism	isoclinic class of AGL(1,𝔽₁₁)
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₁) (order 440) · Inn(G) = AGL(1,𝔽₁₁) · Out(G) = C₂²
homology	Abelianization (H₁) = C₅×C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	(C₁₁⋊C₅)⋊C₈
unambiguous semidirect product	(C₁₁⋊C₈)⋊C₅

	subgroup	G/H
center, Frattini, hypercenter	C₄	AGL(1,𝔽₁₁)
commutator, hypocenter	C₁₁	C₅×C₈
Fitting, cosocle	C₄×C₁₁	C₂×C₅
socle	C₂×C₁₁	C₄×C₅
prime radical	C₂×C₁₁⋊C₅	C₄
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 88

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2,5,2^2,11|a^2=c,c^2=d,e^a=e^10,e^b=e^9

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(440, 1);