C₃²×#4147 #13089452

order	1440
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₃²×#4147

Splittings

Fitting	C₂×C₃²×C₅×Q₈ ⋊ C₂
Hall	C₃²×C₅ ⋊ 2₋¹⁺⁴
Sylow	C₅ ⋊ C₃²×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center	C₂×C₃²	C₂³×D₅
commutator, cosocle	C₂×C₅	C₂⁴×C₃²
Frattini	C₂	C₂³×C₃²×D₅
Fitting	C₂×C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	C₂⁴
hypercenter	C₂×C₃²×Q₈	D₅
hypocenter	C₅	C₃²×2₋¹⁺⁴
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×Q₈

876 conjugacy classes of subgroups; only the 510 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 510 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 — 4 generators on 27 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5769);