order	312
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊(C₃×C₁₃)

Splittings

Fitting, Hall	Q₈×C₁₃ ⋊ C₃
derived, Sylow	Q₈ ⋊ C₃×C₁₃

	subgroup	G/H
center, socle, hypercenter	C₂×C₁₃	A₄
commutator, cosocle, hypocenter	Q₈	C₃×C₁₃
Frattini	C₂	A₄×C₁₃
Fitting	Q₈×C₁₃	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₁₃

14 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 25);