C₂³×D₃ #249

order	48
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = D₃×AGL(3,𝔽₂) (order 8064) · Inn(G) = D₃ · Out(G) = AGL(3,𝔽₂)
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×D₃
unambiguous semidirect product	C₃⋊C₂⁴
unambiguous semidirect product	(C₂×C₃)⋊C₂³

Fitting, socle	C₂³×C₃ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	D₃
commutator, cosocle, hypocenter	C₃	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂³×C₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³

134 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 4 generators on 9 points:

(1,2)(3,4)(5,6)(8,9);(5,6)(7,8,9);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 51);