C₄×Dic₉ #3668

order	144
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = #120×AGL(1,ℤ/9) (order 1728) · Inn(G) = D₉ · Out(G) = C₃×#120
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

direct product	C₄×Dic₉
unambiguous semidirect product	C₉⋊C₄²

C₉ ⋊ C₄²

	subgroup	G/H
center, hypercenter	C₂×C₄	D₉
commutator, hypocenter	C₉	C₄²
Frattini, socle, prime radical	C₂²×C₃	C₂×D₃
Fitting	C₂×C₄×C₉	C₂
cosocle	C₂²×C₉	C₂²
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄

45 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 11);