C₃²×#4131 #13089129

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#4131

Splittings

Hall	C₃²×C₅ ⋊ #131
Sylow	C₅ ⋊ C₃²×#131

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₃²×C₄
Frattini	C₂	C₂²×C₃²×AGL(1,𝔽₅)
Fitting	C₃²×C₅×D₄	C₄
socle	C₂×C₃²×C₅	C₂²×C₄
prime radical	C₃²×#711	C₂
cosocle	Dic₅	C₂³×C₃²
hypercenter	C₃²×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×#131
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×D₄

372 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 204 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)(7,8,9,11,10)(14,20)(17,23)(21,26)(24,27);(4,5,6)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(8,9,10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5446);