C₃×#139787 #72271633

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃×#139787

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

1584 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1584 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

123 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)(13,17,20,23)(14,21)(18,27,19,26)(22,25), (4,5)(6,8,10,11)(7,9)(12,13,14,18,16,20,21,19)(15,23,25,26,24,17,22,27), (4,6)(5,8,9,11)(7,10)(13,18,20,19)(15,24)(17,26,23,27)(22,25));