C₂³×#226 #87478

order	384
isoclinism	isoclinic class of #226
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle, hypercenter	C₂⁴	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴
Frattini	C₂	C₂³×S₄
Fitting	C₂³×Q₈	D₃
prime radical	C₂³×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂⁴

803 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20054);