C₅×#4999 #11750498

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#4999

Splittings

Hall	C₃×C₅ ⋊ #424
Sylow	C₃ ⋊ C₅×#424

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×C₅	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₅×#117	D₃
hypocenter	C₃	C₅×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#117

228 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 28
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5)(6,7,8)(9,10,11,12)(15,21)(18,24)(22,27)(25,28);(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(7,8)(10,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#68774, C₅×#70292, #14300281, …
maximal normal subgroup of	C₅×#68774, C₅×#70292, #14300281, …
maximal quotient of	C₅×#70749, C₅×#70750, C₅×#70751, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4693);