#4246⋊C₃ #92645

order	504
isoclinism	isoclinic class of #4245⋊C₃
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₃×AGL(1,𝔽₇) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Hall	C₃×ΓL(1,𝔽₈) ⋊ D₄
Sylow	C₇ ⋊ C₃×#67

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×AGL(1,𝔽₇)
commutator, Fitting, socle, cosocle	C₂×C₃×C₇	C₂²×C₃
hypocenter	C₃×C₇	C₃×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂

74 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 71);