C₁₁⋊C₈ #761

order	88
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₁) (order 440) · Inn(G) = D₁₁ · Out(G) = C₂²×C₅
holomorph	G ⋊ Aut(G) = #72222646
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

C₁₁⋊C₈

C₁₁ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₁
commutator, hypocenter	C₁₁	C₈
Fitting, cosocle	C₄×C₁₁	C₂
socle, prime radical	C₂×C₁₁	C₄
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₄

8 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:

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

Polycyclic presentation:

2^3,11|a^2=b,b^2=c,d^a=d^10

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(88, 1);