order	154
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₁₁) (order 660) · Inn(G) = D₁₁ · Out(G) = C₂×C₃×C₅
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₁
unambiguous semidirect product	C₁₁⋊(C₂×C₇)

Splittings

Fitting, socle, Hall	C₇×C₁₁ ⋊ C₂
derived, Sylow	C₁₁ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₁	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₇

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 77

A minimal faithful permutation representation — 2 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(154, 2);