D₉×C₂₃ #88023

order	414
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂×C₁₁×AGL(1,ℤ/9) (order 1188) · Inn(G) = D₉ · Out(G) = C₂×C₃×C₁₁
homology	Abelianization (H₁) = C₂×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₉×C₂₃
unambiguous semidirect product	C₉⋊(C₂×C₂₃)

Fitting, Hall	C₉×C₂₃ ⋊ C₂
derived, Sylow	C₉ ⋊ C₂×C₂₃

	subgroup	G/H
center, hypercenter	C₂₃	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₂₃
Frattini	C₃	D₃×C₂₃
Fitting	C₉×C₂₃	C₂
socle	C₃×C₂₃	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂₃

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 207

A minimal faithful permutation representation — 2 generators on 32 points:

(2,4)(3,7)(5,8)(6,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(1,6,4,3,8,5,7,2,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(414, 1);