C₂²×#272 #6276

order	216
isoclinism	isoclinic class of #272
automorphism	Aut(G) = S₄×AGL(2,𝔽₃) (order 10368) · Inn(G) = C₃⋊D₃ · Out(G) = S₄²
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Fitting	C₂²×H(3,𝔽₃) ⋊ C₂
derived, Sylow	H(3,𝔽₃) ⋊ C₂³

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂³
Frattini	C₃	C₂²×C₃⋊D₃
Fitting	C₂²×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

176 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

(1,2)(3,4)(6,8)(9,12)(11,13);(3,4)(5,6,8)(7,9,12)(10,13,11);(1,2)(6,9,13)(8,11,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 113);