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C₃×#329 #4576

order	192
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	Aut(G) = C₂×#23900 (order 512) · Inn(G) = C₂²⋊C₄ · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃×C₄
Frattini, cosocle	C₄²	C₂²×C₃
prime radical	C₂×C₃×D₄	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,6,4,5,7,8)(10,13,14,15)(17,18,19);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67460, #67461, D₃×#329, …
maximal subgroup of	#67460, #67461, C₃×#1403, …
maximal normal subgroup of	#67460, #67461, C₃×#1403, …
maximal quotient of	C₃×#1270, C₃×#1272, C₃×#1295, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 135);