C₂×#4181 #66383

order	324
isoclinism	isoclinic class of #4181
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

Fitting	C₂×C₃²⋊C₉ ⋊ C₂
derived, Sylow	C₃²⋊C₉ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊D₉
commutator, cosocle, hypocenter	C₃²⋊C₉	C₂²
Frattini	C₃²	C₂×C₃⋊D₃
Fitting	C₂×C₃²⋊C₉	C₂
socle	C₂×C₃²	C₃⋊D₃
LCS	G ▷ C₃²⋊C₉
UCS	C₁ ◁ C₂×C₃

99 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 75);