C₅×#5059 #11750558

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#5059

Splittings

Fitting	C₃×C₅×#124 ⋊ C₂
Hall	C₃×C₅ ⋊ #495
Sylow	C₃ ⋊ C₅×#495

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#213	C₂
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#124

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

normal subgroups only

This group has 260 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) = 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)(10,12)(14,19)(15,23)(16,25)(18,26)(20,21)(22,27);(9,10,11,12)(13,15)(14,18)(16,24)(17,23)(19,27)(20,26)(21,22)(25,28);(7,8)(10,12)(13,14)(15,22)(16,19)(17,21)(18,23)(20,25)(24,26)(27,28)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#495, C₂×C₅×#5059, D₅×#5059, …
maximal normal subgroup of	C₅×D₃×#495, C₂×C₅×#5059, D₅×#5059, …
maximal quotient of	C₅×#71180, C₅×#71212, C₅×#71181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4753);