C₇×GL(2,𝔽₃) #66647

order	336
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂²×C₃×S₄ (order 288) · Inn(G) = S₄ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

Fitting	C₇×Q₈ ⋊ D₃
derived	SL(2,𝔽₃) ⋊ C₂×C₇

	subgroup	G/H
center, socle, hypercenter	C₂×C₇	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂×C₇
Frattini	C₂	C₇×S₄
Fitting	C₇×Q₈	D₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 116);