C₂×#1113 #6335

order	216
isoclinism	isoclinic class of #1113
automorphism	Aut(G) = #72276300 (order 5184) · Inn(G) = #1113 · Out(G) = C₂×S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Fitting, socle	C₂×C₃³ ⋊ C₂²
derived, Sylow	C₃³ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂	#1113
commutator, cosocle, hypocenter	C₃³	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃³	C₂²
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

162 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 172);