C₃×#147817 #72254286

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

Aliases

direct product

C₃×#147817

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1790
commutator	C₂²×D₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×#41	C₂³×C₃
prime radical	C₃×#37269	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

2412 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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#147817, #72270026
maximal subgroup of	C₃×#72215601, D₃×#147817, #72270026
maximal normal subgroup of	C₃×#72215601, D₃×#147817, #72270026

GAP

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

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

C3×#147817 #72254286