A₄⋊(ΓL(1,𝔽₈)⋊C₄) #11759352

order	1008
isoclinism	isoclinic class of A₄⋊AGL(1,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	A₄⋊(ΓL(1,𝔽₈)⋊C₄)
unambiguous semidirect product	ΓL(1,𝔽₈)⋊SL(2,ℤ/4)
unambiguous semidirect product	(C₇⋊SL(2,ℤ/4))⋊C₃

derived	C₇×A₄ ⋊ C₃×C₄
Sylow	C₇ ⋊ C₃×SL(2,ℤ/4)

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄⋊AGL(1,𝔽₇)
commutator, hypocenter	C₇×A₄	C₃×C₄
Fitting, socle	C₂³×C₇	C₃×D₃
prime radical	C₂×A₄×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇×A₄	C₂×C₃
LCS	G ▷ C₇×A₄
UCS	C₁ ◁ C₂

84 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 523);