D₇⋊SL(2,𝔽₃) #66666

order	336
isoclinism	isoclinic class of #66665
automorphism	Aut(G) = C₂×A₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₇⋊A₄ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇⋊A₄
commutator, Fitting, cosocle, hypocenter	C₇×Q₈	C₂×C₃
socle	C₂×C₇	C₂×A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂

36 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:

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

Polycyclic presentation:

2,3,2^2,7,2|c^2=f,d^2=f,e^a=e^6,c^b=d,d^b=cd,e^b=e^4,d^c=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 135);