C₂³×D₉ #3769

order	144
isoclinism	isoclinic class of D₉
automorphism	Aut(G) order 72576
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×D₉
unambiguous semidirect product	C₉⋊C₂⁴
unambiguous semidirect product	(C₂×C₉)⋊C₂³

Fitting	C₂³×C₉ ⋊ C₂
derived, Sylow	C₉ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	D₉
commutator, cosocle, hypocenter	C₉	C₂⁴
Frattini	C₃	C₂³×D₃
Fitting	C₂³×C₉	C₂
socle	C₂³×C₃	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂³

201 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 4 generators on 15 points:

(1,2)(3,4)(5,6)(8,10)(9,13)(11,14)(12,15);(5,6)(7,12,10,9,14,11,13,8,15);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 112);