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C₃×#423 #5297

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂²×#511790 (order 2048) · Inn(G) = C₂³ · Out(G) = C₂³×#120
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Subgroups

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

126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67710, #67711, C₃⋊#1309, …
maximal subgroup of	#67710, #67711, C₃⋊#1309, …
maximal normal subgroup of	#67710, #67711, C₃⋊#1309, …
maximal quotient of	C₃×#1566, C₃×#1567, C₃×#1568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 856);