Q₈⋊C₉ #606

order	72
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₃×S₄ (order 72) · Inn(G) = A₄ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72219864
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

Q₈⋊C₉

Q₈ ⋊ C₉

	subgroup	G/H
center, Frattini, socle, prime radical, hypercenter	C₂×C₃	A₄
commutator, hypocenter	Q₈	C₉
Fitting, cosocle	C₃×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃

11 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

3^2,2^3|a^3=b,c^2=e,d^2=e,c^a=d,d^a=cd,d^c=de

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 3);