order	242
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂×C₅×AGL(1,𝔽₁₁) (order 1100) · Inn(G) = D₁₁ · Out(G) = C₂×C₅²
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

affine special semilinear	AΣL(1,𝔽₁₂₁)
direct product	C₁₁×D₁₁
unambiguous semidirect product	C₁₁⋊(C₂×C₁₁)

Splittings

Fitting, socle, Sylow	C₁₁² ⋊ C₂
derived	C₁₁ ⋊ C₂×C₁₁

	subgroup	G/H
center, hypercenter	C₁₁	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₁₁²	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₁₁

13 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 22

A minimal faithful permutation representation — 2 generators on 22 points:

(1,6,5,14,13,21,20,19,12,11,4,3,2,10,9,18,17,22,16,15,8,7);(1,4,8,12,16,20,17,13,9,5,2)(3,6,10,14,18,21,22,19,15,11,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(242, 3);