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C₃×AΣL(1,𝔽₁₆) #4598

order	192
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	Aut(G) = C₂²×B(4,𝔽₂) (order 256) · Inn(G) = #99 · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₃×AΣL(1,𝔽₁₆)

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#99
commutator	C₂³	C₂×C₃×C₄
Frattini, cosocle	C₂×D₄	C₂²×C₃
prime radical	C₃×#120	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂×C₃×D₄ ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#69130, #69131, #69132, …
maximal subgroup of	#69130, #69131, #69132, …
maximal normal subgroup of	#69130, #69131, #69132, …
maximal quotient of	C₃×#1316, C₃×#1326, C₃×#1343, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 157);