C₅×#71250 #14290976

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

Aliases

direct product

C₅×#71250

Splittings

Hall	C₃×C₅ ⋊ #1658
Sylow	C₃ ⋊ C₅×#1658

Subgroups

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#490	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₃×C₅×#127	C₂²
hypercenter	C₅×#490	D₃
hypocenter	C₃	C₅×#1658
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#490

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

normal subgroups only

This group has 364 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)(6,7,8)(9,11,13,18)(10,15,16,24)(12,21,19,17)(14,22,20,23)(25,27)(26,30)(28,37)(29,34)(31,40)(32,39)(33,35)(36,38);(7,8)(9,10,13,16)(11,15,18,24)(12,20,19,14)(17,22,21,23)(25,26,29,32)(27,30,34,39)(28,36,35,31)(33,40,37,38);(10,14)(12,19)(15,23)(16,20)(17,21)(22,24)(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, 16674);