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C₂×C₃×M₁₆ #970

order	96
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×B(4,𝔽₂) (order 256) · Inn(G) = C₂² · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₂×C₃×M₁₆

Subgroups

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

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10682675, #10682676, #10682677, …
Frattini subgroup of	C₉×#1320, C₉×#1321, C₉×#1327, …
Fitting subgroup of	#4553, #4554, #4555, …
maximal subgroup of	#4553, #4554, #4555, …
maximal normal subgroup of	#4553, #4554, #4555, …
maximal quotient of	C₂×C₃×#97, C₃×C₄×M₁₆, C₂×C₃×C₂²⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 177);