#41⋊ΓL(1,𝔽₈) #66704

order	336
isoclinism	isoclinic class of C₇⋊SL(2,𝔽₃)
automorphism	Aut(G) = C₂×A₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = C₂²⋊ΓL(1,𝔽₈) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, hypercenter	C₄	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇×Q₈	C₂×C₃
Frattini	C₂	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#41	C₃
socle	C₂×C₇	C₂×A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₄

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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 173);