C₃×#147577 #72254285

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

Aliases

direct product

C₃×#147577

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1781
commutator	C₂³×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×#41	C₂³×C₃
prime radical	C₃×#35527	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×H(5,𝔽₂) ◁ G

3924 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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#147577, #72269995
maximal subgroup of	C₃×#72215601, D₃×#147577, #72269995
maximal normal subgroup of	C₃×#72215601, D₃×#147577, #72269995

GAP

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

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

C3×#147577 #72254285