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C₃×#339 #4586

order	192
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂²×#3484 (order 512) · Inn(G) = D₄ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	D₄
commutator	C₄	C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂²×C₃
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×C₃	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

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

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#68728, #68729, #68730, …
maximal subgroup of	#68728, #68729, #68730, …
maximal normal subgroup of	#68728, #68729, #68730, …
maximal quotient of	C₃×#1276, C₃×#1278, C₃×#1279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 145);