C₃²×#4096 #13087875

order	1440
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#4096

Splittings

Fitting	C₃²×C₅×#41 ⋊ C₂
Hall	C₃²×C₅ ⋊ #135
Sylow	C₅ ⋊ C₃²×#135

Subgroups

	subgroup	G/H
center	C₃²×C₄	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³×C₃²
Frattini	C₄	C₂²×C₃²×D₅
Fitting	C₃²×C₅×#41	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₃²×#41	D₅
hypocenter	C₅	C₃²×#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#41

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

normal subgroups only

This group has 372 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)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(4,5,6)(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(8,9)(10,11)(12,13)(14,17)(15,18)(16,20)(19,23)(21,24)(22,26)(25,27)

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, 4192);