(C₃⋊AGL(1,𝔽₇))⋊C₄ #92643

order	504
isoclinism	isoclinic class of D₃×AGL(1,𝔽₇)
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₃×AGL(1,𝔽₇) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	(C₃⋊AGL(1,𝔽₇))⋊C₄
unambiguous semidirect product	(D₂₁⋊C₄)⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×AGL(1,𝔽₇)
commutator, hypocenter	C₃×C₇	C₂×C₃×C₄
Fitting, socle, cosocle	C₂×C₃×C₇	C₂²×C₃
prime radical	C₂×C₃⋊AGL(1,𝔽₇)	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂

70 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)(4,6,7,5)(8,9,14)(10,11,13);(1,2,3)(4,5,7,6)(9,10)(11,13)(12,14)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 69);