C₃×#146479 #72271312

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#146479

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1781
commutator	C₂²×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂³×D₄	C₂³×C₃
prime radical	C₂⁴×C₃×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃×D₄ ◁ G

4952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

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

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#146479, #72271349, #72271454
maximal subgroup of	C₃×#72219159, D₃×#146479, #72271349, …
maximal normal subgroup of	C₃×#72219159, D₃×#146479, #72271349, …

GAP

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

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

C3×#146479 #72271312