C₃⋊(D₇⋊A₄) #92760

order	504
isoclinism	isoclinic class of C₃⋊(D₇⋊A₄)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₃⋊(D₇⋊A₄)
unambiguous semidirect product	D₂₁⋊A₄
unambiguous semidirect product	(C₂²⋊ΓL(1,𝔽₈))⋊D₃

Fitting, socle, derived	C₂²×C₃×C₇ ⋊ C₂×C₃
Sylow	C₇ ⋊ D₃×A₄

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

52 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 186);