Dic₇⋊A₄ #66667

order	336
isoclinism	isoclinic class of D₇⋊A₄
automorphism	Aut(G) = C₂×A₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = D₇⋊A₄ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	Dic₇⋊A₄
unambiguous semidirect product	(C₂²⋊ΓL(1,𝔽₈))⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇⋊A₄
commutator, hypocenter	C₂²×C₇	C₃×C₄
Fitting, socle, cosocle	C₂³×C₇	C₂×C₃
prime radical	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂

38 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 15 points:

(2,6,5,3,4,7)(8,9,10,11)(13,14,15);(1,2,5,6,4,7,3)(12,15)(13,14)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 136);