C₃×D₃×ΓL(1,𝔽₈) #67395

order	378
isoclinism	isoclinic class of D₃×ΓL(1,𝔽₈)
automorphism	Aut(G) = D₃²×AGL(1,𝔽₇) (order 1512) · Inn(G) = D₃×ΓL(1,𝔽₈) · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×D₃×ΓL(1,𝔽₈)
unambiguous semidirect product	C₃⋊(C₂×C₃×ΓL(1,𝔽₈))
unambiguous semidirect product	(D₃×C₇)⋊C₃²
unambiguous semidirect product	(C₃×D₃×C₇)⋊C₃

	subgroup	G/H
center, hypercenter	C₃	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

(5,6)(8,9,10)(11,13,12);(1,2,3)(4,5,6)(7,8,9,11,10,12,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 48);