C₇⋊SL(2,𝔽₃) #4253

order	168
isoclinism	isoclinic class of C₇⋊SL(2,𝔽₃)
automorphism	Aut(G) = A₄×AGL(1,𝔽₇) (order 504) · Inn(G) = C₂²⋊ΓL(1,𝔽₈) · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₇⋊SL(2,𝔽₃)
unambiguous semidirect product	Q₈⋊ΓL(1,𝔽₈)

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

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

Polycyclic presentation:

3,2^2,7,2|b^2=e,c^2=e,b^a=c,c^a=bc,d^a=d^2,c^b=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 23);