C₃×#139710 #72271638

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#139710

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1889
commutator	C₂²×Q₈	C₂²×C₃×C₄
Frattini, cosocle	C₂³×Q₈	C₂³×C₃
prime radical	C₃×#23910	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×Q₈ ◁ G

1724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

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

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

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72213489
maximal normal subgroup of	C₃×#72213489

GAP

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

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

C3×#139710 #72271638