order	154
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₅×AGL(1,𝔽₇) (order 420) · Inn(G) = D₇ · Out(G) = C₂×C₃×C₅
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₁)×#64423⋊C₂
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁×D₇
unambiguous semidirect product	C₇⋊(C₂×C₁₁)

Splittings

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

	subgroup	G/H
center, hypercenter	C₁₁	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₇×C₁₁	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₁

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(154, 1);