C₃×#493480 #72272032

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₃×#493480

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#20585
commutator	C₂²×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴×C₃
prime radical	C₃×#62832	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₃×#3484 ◁ G

5234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72272045, D₃×#493480
maximal subgroup of	#72272045, D₃×#493480, C₃×#72216366
maximal normal subgroup of	#72272045, D₃×#493480, C₃×#72216366

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

C3×#493480 #72272032