order	138
isoclinism	isoclinic class of D₂₃
automorphism	Aut(G) = C₂×AGL(1,𝔽₂₃) (order 1012) · Inn(G) = D₂₃ · Out(G) = C₂×C₁₁
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₃
unambiguous semidirect product	C₂₃⋊(C₂×C₃)

Splittings

Fitting, socle, Hall	C₃×C₂₃ ⋊ C₂
derived, Sylow	C₂₃ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₂₃	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₃

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 69

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26);(4,5,7,9,11,13,15,17,19,21,23,25,26,24,22,20,18,16,14,12,10,8,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(138, 2);