C₃×#501097 #72270048

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#501097

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴×C₃
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

3084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#501097, #72270771
maximal subgroup of	C₃×#72210575, D₃×#501097, #72270771
maximal normal subgroup of	C₃×#72210575, D₃×#501097, #72270771

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

C3×#501097 #72270048