C₅×#64110 #13088084

order	1440
isoclinism	isoclinic class of #634
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₅×#64110

Splittings

Fitting	C₂×C₃²×C₅×C₈ ⋊ C₂
Hall	C₃²×C₅ ⋊ #131
Sylow	C₃² ⋊ C₅×#131

Subgroups

	subgroup	G/H
center	C₅×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×C₃⋊D₃
Fitting	C₂×C₃²×C₅×C₈	C₂
socle	C₂×C₃²×C₅	C₂²×C₄
prime radical	C₅×#3828	C₂
cosocle	C₃²×C₄	C₂³×C₅
hypercenter	C₂×C₅×C₈	C₃⋊D₃
hypocenter	C₃²	C₅×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈

372 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4401);