C₅×#5204 #11750703

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#5204

Splittings

Hall	C₃×C₅ ⋊ #473
Sylow	C₃ ⋊ C₅×#473

Subgroups

	subgroup	G/H
center	C₂×C₅	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#137	C₂
socle	C₂×C₃×C₅	#121
prime radical	C₃×C₅×#41	C₂²
hypercenter	C₅×#137	D₃
hypocenter	C₃	C₅×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#137

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7,8)(9,12,15,29)(10,17,20,35)(11,22,24,38)(13,28,14,27)(16,31,32,40)(18,34,19,25)(21,37,36,33)(23,39,30,26);(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,33)(18,21)(19,36)(20,32)(22,28)(25,40)(27,38)(29,39)(31,34)(35,37);(7,8)(9,10,15,20)(11,21,24,36)(12,25,29,34)(13,19,14,18)(16,30,32,23)(17,28,35,27)(22,31,38,40)(26,37,39,33)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#473, C₂×C₅×#5204, D₅×#5204, …
maximal normal subgroup of	C₅×D₃×#473, C₂×C₅×#5204, D₅×#5204, …
maximal quotient of	C₅×#71101, C₅×#71117, C₅×#71037, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4898);