C₃×#146654 #72248304

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#146654

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1790
commutator	C₂×D₄	C₂×C₃×C₄²
Frattini, cosocle	C₂³×D₄	C₂³×C₃
prime radical	C₂×C₃×#535	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

2434 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:

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#146654, #72248514
maximal subgroup of	D₃×#146654, #72248514, C₃×#72213329
maximal normal subgroup of	D₃×#146654, #72248514, C₃×#72213329
maximal quotient of	C₃×#72210462

GAP

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

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

C3×#146654 #72248304