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C₃×#343 #4590

order	192
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	Aut(G) = C₂×#511771 (order 1024) · Inn(G) = C₂²⋊C₄ · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂²⋊C₄
commutator	C₂²	C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂²×C₃
prime radical	C₂³×C₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67493, #67494, #67495, …
maximal subgroup of	#67493, #67494, #67495, …
maximal normal subgroup of	#67493, #67494, #67495, …
maximal quotient of	C₃×#1280, C₃×#1283, C₃×#1285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 149);