Dic₁₁⋊C₅ #6346

order	220
isoclinism	isoclinic class of AGL(1,𝔽₁₁)
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₁) (order 220) · Inn(G) = AGL(1,𝔽₁₁) · Out(G) = C₂
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	Dic₁₁⋊C₅
unambiguous semidirect product	(C₁₁⋊C₅)⋊C₄

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

12 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 1);