C₃×#491469 #72225816

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₃×#491469

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 27 points:

(2,5)(3,6)(7,8)(10,14)(11,18)(12,19)(15,21)(16,20)(17,24)(22,23)(25,26,27);(10,15)(14,21)(17,22)(23,24);(1,2,6,8,4,7,3,5);(9,10,16,24,12,17,11,14,13,15,18,23,19,22,20,21)

Conjugacy classes & character table

276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#72213983, #72225906, D₃×#491469, …
maximal normal subgroup of	C₃×#72213983, #72225906, D₃×#491469, …
maximal quotient of	C₃×#72213956

GAP

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

gap> G := Group((2,5)(3,6)(7,8)(10,14)(11,18)(12,19)(15,21)(16,20)(17,24)(22,23)(25,26,27), (10,15)(14,21)(17,22)(23,24), (1,2,6,8,4,7,3,5), (9,10,16,24,12,17,11,14,13,15,18,23,19,22,20,21));

C3×#491469 #72225816