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C₉×M₁₆ #3681

order	144
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₃×D₄ (order 96) · Inn(G) = C₂² · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

C₉×M₁₆

Subgroups

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

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#63381, #63382, #63383, …
maximal subgroup of	#63381, #63382, #63383, …
maximal normal subgroup of	#63381, #63382, #63383, …
maximal quotient of	C₉×#97, C₉×C₂²⋊C₈, C₉×C₄⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 24);