C₃×#147567 #72248190

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

Aliases

direct product

C₃×#147567

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1781
commutator	C₂⁵	C₂²×C₃×C₄
Frattini, cosocle	C₂³×D₄	C₂³×C₃
prime radical	C₃×#34980	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃×D₄ ◁ G

4902 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)(4,5,6,7)(8,9)(10,16,17,23)(11,19)(12,14)(13,18,22,20)(15,21);(8,10)(9,13)(11,20,21,18)(12,17)(14,22)(15,23,19,16);(5,7)(11,21)(13,22)(15,19)(18,20)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#147567, #72248464
maximal subgroup of	D₃×#147567, #72248464, C₃×#72213329
maximal normal subgroup of	D₃×#147567, #72248464, C₃×#72213329
maximal quotient of	C₃×#72213256

GAP

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

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

C3×#147567 #72248190