C₂×C₃×SL(2,𝔽₃) #3813

order	144
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂×D₃×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₂×C₃×SL(2,𝔽₃)
unambiguous semidirect product	(C₂×Q₈)⋊C₃²
unambiguous semidirect product	(C₂×C₃×Q₈)⋊C₃

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃²
Frattini	C₂	C₂×C₃×A₄
Fitting	C₂×C₃×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₃

50 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 156);