C₅×#71253 #14291214

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#71253

Splittings

Fitting	C₃×C₅×#491 ⋊ C₂
Hall	C₃×C₅ ⋊ #1664
Sylow	C₃ ⋊ C₅×#1664

Subgroups

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#491	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₅×D₁₂	D₄
hypercenter	C₅×#491	D₃
hypocenter	C₃	C₅×#1664
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#491

368 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 40
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(25,27,29,34)(26,30,32,39)(28,37,35,33)(31,40,36,38);(6,7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,19)(18,23)(25,26,29,32)(27,30,34,39)(28,36,35,31)(33,40,37,38);(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20)(27,33)(28,35)(30,38)(31,36)(34,37)(39,40)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 16912);