Finite Group Databasebeta

← Prev (#5249) Next (#5251) →

C₃×#376 #5250

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

Subgroups

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

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67614, #67615, #69430, …
maximal subgroup of	#67614, #67615, C₃×#1274, …
maximal normal subgroup of	#67614, #67615, C₃×#1274, …
maximal quotient of	C₃×C₄×#95, C₃×#1434, C₃×#1744, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 809);