order	238
isoclinism	isoclinic class of D₁₇
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₁₇) (order 1632) · Inn(G) = D₁₇ · Out(G) = C₂×C₃×C₈
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₇
unambiguous semidirect product	C₁₇⋊(C₂×C₇)

Splittings

Fitting, socle, Hall	C₇×C₁₇ ⋊ C₂
derived, Sylow	C₁₇ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₇

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 119

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);(8,9,11,13,15,17,19,21,23,24,22,20,18,16,14,12,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(238, 2);