Q₈×#226 #85427

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, Frattini, socle	C₂²	C₂²×S₄
commutator, prime radical, cosocle	C₂×SL(2,𝔽₃)	C₂³
Fitting	Q₈²	D₃
hypercenter	C₂×Q₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

195 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18003);