C₃²×#4025 #13087805

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#4025

Splittings

Hall	C₃²×C₅ ⋊ #126
Sylow	C₅ ⋊ C₃²×#126

Subgroups

	subgroup	G/H
center	C₂²×C₃²	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₃²
Frattini	C₂²	C₂²×C₃²×D₅
Fitting	C₃²×C₅×C₂²⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂³×C₃²×C₅	C₂²
hypercenter	C₃²×C₂²⋊C₄	D₅
hypocenter	C₅	C₃²×#126
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₂²⋊C₄

360 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 720

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4122);