C₃×#494600 #72248348

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#494600

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴×C₃
prime radical	C₂×C₃×#3019	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

4346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72248361, D₃×#494600
maximal subgroup of	#72248361, D₃×#494600, C₃×#72213329
maximal normal subgroup of	#72248361, D₃×#494600, C₃×#72213329
maximal quotient of	C₃×#72210479

GAP

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

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

C3×#494600 #72248348