C₇⋊(A₄⋊C₈) #10620701

order	672
isoclinism	isoclinic class of C₇⋊S₄
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₇⋊(A₄⋊C₈)
unambiguous semidirect product	A₄⋊(C₇⋊C₈)

derived	C₇×A₄ ⋊ C₈
Sylow	C₇ ⋊ A₄⋊C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₇⋊S₄
commutator, hypocenter	C₇×A₄	C₈
Fitting	C₂²×C₄×C₇	D₃
socle	C₂³×C₇	Dic₃
prime radical	C₂×C₇×A₄	C₄
cosocle	C₄×C₇×A₄	C₂
LCS	G ▷ C₇×A₄
UCS	C₁ ◁ C₄

52 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 168

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 283);