D₇⋊(C₂²⋊C₉) #92632

order	504
isoclinism	isoclinic class of D₇⋊A₄
automorphism	Aut(G) = C₃×A₄×AGL(1,𝔽₇) (order 1512) · Inn(G) = D₇⋊A₄ · Out(G) = C₃²
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₇⋊(C₂²⋊C₉)
unambiguous semidirect product	(C₂²⋊(C₇⋊C₉))⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₇⋊A₄
commutator, hypocenter	C₂²×C₇	C₂×C₉
Fitting, socle, cosocle	C₂²×C₃×C₇	C₂×C₃
prime radical	C₂²×C₃×D₇	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 20 points:

(2,6,5,3,4,7)(9,10,11)(12,16,20,14,19,15,17,13,18);(1,2,5,6,4,7,3)(8,11)(9,10)

Polycyclic presentation:

2,3^2,2^2,7|b^3=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(504, 58);