C₇⋊(Q₈⋊C₉) #92597

order	504
isoclinism	isoclinic class of C₇⋊SL(2,𝔽₃)
automorphism	Aut(G) = C₃×A₄×AGL(1,𝔽₇) (order 1512) · Inn(G) = C₂²⋊ΓL(1,𝔽₈) · Out(G) = C₂×C₃²
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₇⋊(Q₈⋊C₉)
unambiguous semidirect product	Q₈⋊(C₇⋊C₉)

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₇×Q₈	C₉
Fitting, cosocle	C₃×C₇×Q₈	C₃
socle, prime radical	C₂×C₃×C₇	A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂×C₃

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 24 points:

(2,3,4)(5,7,6)(9,10,12)(13,15,14)(16,17,19,18,20,22,21,23,24);(1,2,3,5,4,6,7)(8,12,11,14)(9,15,13,10)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 23);