C₂²×#226 #5915

order	192
isoclinism	isoclinic class of #226
automorphism	Aut(G) = S₄×#5934 (order 4608) · Inn(G) = S₄ · Out(G) = #5934
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

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

153 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1474);