C₃×#364468 #72269872

order	1536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₃×#364468

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#386
commutator	C₂²×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#3003	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

5860 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)(8,9,11,15,13,17,12,16)(10,18,20,23,19,22,21,14);(4,5)(6,7)(8,10,11,21,13,19,12,20)(9,14,16,18,17,23,15,22);(5,7);(9,15)(10,19)(11,12)(14,22)(16,17)(18,23)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#364468, #72271445
maximal subgroup of	D₃×#364468, #72271445, C₃×#72215520
maximal normal subgroup of	D₃×#364468, #72271445, C₃×#72215520

GAP

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

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

C3×#364468 #72269872