C₅×#5200 #11750699

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

Aliases

direct product

C₅×#5200

Splittings

Hall	C₃×C₅ ⋊ #471
Sylow	C₃ ⋊ C₅×#471

Subgroups

	subgroup	G/H
center	C₂×C₅	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×AGL(1,ℤ/8)	C₂
socle	C₂×C₃×C₅	#121
hypercenter	C₅×AGL(1,ℤ/8)	D₃
hypocenter	C₃	C₅×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×AGL(1,ℤ/8)

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

normal subgroups only

This group has 208 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#471, C₂×C₅×#5200, D₅×#5200, …
maximal normal subgroup of	C₅×D₃×#471, C₂×C₅×#5200, D₅×#5200, …
maximal quotient of	C₅×#71004, C₅×#71020, C₅×#71132, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4894);