Dic₉⋊C₉ #66317

order	324
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	Aut(G) = C₂×C₃×AGL(1,ℤ/9) (order 324) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	Dic₉⋊C₉
unambiguous semidirect product	(C₉⋊C₉)⋊C₄

derived	C₉ ⋊ C₄×C₉
Sylow	C₉⋊C₉ ⋊ C₄

	subgroup	G/H
center, hypercenter	C₂×C₃	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₄×C₉
Frattini, socle, prime radical	C₂×C₃²	C₃×D₃
Fitting	C₂×C₉⋊C₉	C₂
cosocle	C₂×C₃×C₉	C₂×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₃

35 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:

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

Polycyclic presentation:

2,3,2,3^3|a^2=c,b^3=d,e^3=f^2,e^a=e^2f,f^a=f^2,e^b=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 9);