C₂³×#41⋊C₃ #87529

order	384
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, hypercenter	C₂³×C₄	A₄
commutator, cosocle, hypocenter	Q₈	C₂⁴×C₃
Frattini	C₂	C₂⁴×A₄
Fitting	C₂³×#41	C₃
socle	C₂⁴	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³×C₄

1395 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20105);