C₇×#226 #66646

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

	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
prime radical	C₇×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(1,15,10,7,5,6,12,13)(2,3,14,9,8,11,16,4)(17,18,19,20,21,22,23);(3,9,12)(4,10,11)(6,13,16)(7,14,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 115);