Dic₂₂ #763

order	88
isoclinism	isoclinic class of Dic₂₂
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₁) (order 880) · Inn(G) = C₂×D₁₁ · Out(G) = C₂²×C₅
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₂
unambiguous semidirect product	C₁₁⋊Q₈

C₁₁ ⋊ Q₈

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₁
commutator, socle, prime radical, cosocle	C₂×C₁₁	C₂²
Fitting	C₄×C₁₁	C₂
hypercenter	C₄	D₁₁
hypocenter	C₁₁	Q₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₄

12 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(88, 3);