C₅×#69316 #14318199

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#69316

Splittings

Fitting	C₃×C₅×#365 ⋊ C₂
Hall	C₃×C₅ ⋊ #2231
Sylow	C₃ ⋊ C₅×#2231

Subgroups

	subgroup	G/H
center	C₂×C₅	#4870
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#365	C₂
socle	C₂×C₃×C₅	#476
prime radical	C₅×#904	C₂²
hypercenter	C₅×#365	D₃
hypocenter	C₃	C₅×#2231
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#365

240 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 480

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

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 43897);