Q₈⋊C₂₇ #6166

order	216
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₉×S₄ (order 216) · Inn(G) = A₄ · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₂₇ · Schur multiplier (H₂) = C₁

Aliases

Q₈⋊C₂₇

Q₈ ⋊ C₂₇

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₉	A₄
commutator, hypocenter	Q₈	C₂₇
Fitting, cosocle	Q₈×C₉	C₃
socle, prime radical	C₂×C₃	C₂²⋊C₉
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₉

15 conjugacy classes of subgroups.

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 2 generators on 35 points:

(2,3,5)(6,8,7)(9,10,13,11,14,19,16,21,26,12,15,20,17,22,27,24,29,32,18,23,28,25,30,33,31,34,35);(1,5,4,7)(2,8,6,3)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 3);