C₉×Dic₉ #66314

order	324
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂²×C₃×AGL(1,ℤ/9) (order 648) · Inn(G) = D₉ · Out(G) = C₂²×C₃²
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

derived	C₉ ⋊ C₄×C₉
Sylow	C₉² ⋊ C₄

	subgroup	G/H
center, hypercenter	C₂×C₉	D₉
commutator, hypocenter	C₉	C₄×C₉
Frattini, socle, prime radical	C₂×C₃²	C₃×D₃
Fitting	C₂×C₉²	C₂
cosocle	C₂×C₃×C₉	C₂×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₉

41 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 6);