C₃×#139777 #72253574

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#139777

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#1889
commutator	C₂²×D₄	C₂²×C₃×C₄
Frattini, cosocle	C₂³×D₄	C₂³×C₃
prime radical	C₃×#3484	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

1780 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1780 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)(4,5)(6,10,11,8)(7,9)(12,13)(14,20,22,17)(15,24)(16,18)(19,25)(21,27,23,26);(4,6)(5,8)(7,11)(9,10)(12,14)(13,17)(15,21)(16,22)(18,20)(19,26)(23,25)(24,27);(4,5)(6,10,11,8)(7,9)(14,21)(17,26)(20,27)(22,23)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#139777, #72270245
maximal subgroup of	C₃×#72210479, D₃×#139777, #72270245
maximal normal subgroup of	C₃×#72210479, D₃×#139777, #72270245
maximal quotient of	C₃×#72216507

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