C₃²×#4088 #13087869

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#4088

Splittings

Fitting	C₂×C₃²×C₅×Q₈ ⋊ C₂
Hall	C₃²×C₅ ⋊ #137
Sylow	C₅ ⋊ C₃²×#137

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³×C₃²
Frattini	C₄	C₂²×C₃²×D₅
Fitting	C₂×C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	C₂×D₄
prime radical	C₃²×#709	C₂
hypercenter	C₂×C₃²×Q₈	D₅
hypocenter	C₅	C₃²×#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×Q₈

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4186);