C₃×#141754 #72225866

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₃×#141754

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1957
commutator	C₄×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³×C₃
prime radical	C₂×C₃×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72213983, C₃×#72214020
maximal normal subgroup of	C₃×#72213983, C₃×#72214020
maximal quotient of	C₃×#72214020

GAP

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

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

C3×#141754 #72225866