GU(2,𝔽₃) #860

order	96
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂²×S₄ (order 96) · Inn(G) = S₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

GU(2,𝔽₃)

	subgroup	G/H
center, Frattini, hypercenter	C₄	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₄
Fitting	#41	D₃
socle	C₂	SL(2,ℤ/4)
prime radical, cosocle	#41⋊C₃	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄

24 conjugacy classes of subgroups.

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

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 67);