C₇×#41⋊C₃ #66701

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	C₇×#41⋊C₃
unambiguous semidirect product	#41⋊(C₃×C₇)

Fitting, Hall	C₇×#41 ⋊ C₃
Sylow	#41 ⋊ C₃×C₇

	subgroup	G/H
center, hypercenter	C₄×C₇	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃×C₇
Frattini	C₂	C₂×C₇×A₄
Fitting	C₇×#41	C₃
socle	C₂×C₇	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₄×C₇

30 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 170);