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C₃×#353 #4600

order	192
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	Aut(G) = C₂×#33551 (order 512) · Inn(G) = #99 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#99
commutator	C₂×C₄	C₂×C₃×C₄
Frattini, cosocle	C₂×D₄	C₂²×C₃
prime radical	C₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂×C₃×D₄ ◁ G

86 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)(3,6,8,5)(4,7)(9,10,11);(2,5)(3,8)(6,7)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67538, #67539, #67540, …
maximal subgroup of	#67538, #67539, #67540, …
maximal normal subgroup of	#67538, #67539, #67540, …
maximal quotient of	C₃×#1318, C₃×#1325, C₃×#1343, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 159);