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C₉×#41 #3707

order	144
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₃×S₄ (order 288) · Inn(G) = C₂² · Out(G) = C₂²×C₃×D₃
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

Subgroups

	subgroup	G/H
center	C₄×C₉	C₂²
commutator	C₂	C₂³×C₉
Frattini, socle, cosocle	C₂×C₃	C₂³×C₃
prime radical	C₃×#41	C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ G

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13147136, #13147137, #13147142, …
Fitting subgroup of	#63396, #63510, #63511, …
maximal subgroup of	#63396, C₉×#104, #63510, …
maximal normal subgroup of	#63396, C₉×#104, #63510, …
maximal quotient of	C₉×#117, C₄×D₄×C₉, C₄×Q₈×C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 50);