C₂×C₇×SL(2,𝔽₃) #66700

order	336
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂²×C₃×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₇×SL(2,𝔽₃)
unambiguous semidirect product	(C₂×Q₈)⋊(C₃×C₇)

	subgroup	G/H
center, socle, hypercenter	C₂²×C₇	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃×C₇
Frattini	C₂	C₂×C₇×A₄
Fitting	C₂×C₇×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₇

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 169);