C₂×#272 #1101

order	108
isoclinism	isoclinic class of #272
automorphism	Aut(G) = C₂×AGL(2,𝔽₃) (order 864) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×S₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

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

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

55 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 28);