C₃×#147782 #72254120

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#147782

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	B(4,𝔽₂)
commutator	C₂²×D₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×#41	C₂³×C₃
socle	C₂²×C₃	#1790
prime radical	C₃×#52163	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

2412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2)(3,4)(6,11,15,17)(7,16)(9,18)(12,20,14,13)(21,22,23);(5,6,9,14,10,15,18,12)(7,17,8,13,16,11,19,20);(2,4)(6,12)(11,20)(13,17)(14,15)

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72254126, D₃×#147782, #72270941
maximal subgroup of	#72254126, D₃×#147782, C₃×#72215601, …
maximal normal subgroup of	#72254126, D₃×#147782, C₃×#72215601, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(3,4)(6,11,15,17)(7,16)(9,18)(12,20,14,13)(21,22,23), (5,6,9,14,10,15,18,12)(7,17,8,13,16,11,19,20), (2,4)(6,12)(11,20)(13,17)(14,15));

C3×#147782 #72254120