C₃⋊(C₇⋊S₄) #92751

order	504
isoclinism	isoclinic class of C₃⋊(C₇⋊S₄)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃

Aliases

unambiguous semidirect product	C₃⋊(C₇⋊S₄)
unambiguous semidirect product	C₇⋊(C₃⋊S₄)
unambiguous semidirect product	A₄⋊D₂₁

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₇×A₄	C₂
Fitting, socle	C₂²×C₃×C₇	D₃
LCS	G ▷ C₃×C₇×A₄
UCS	C₁

60 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 177);