C₅×#5051 #11750550

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

Aliases

direct product

C₅×#5051

Splittings

Hall	C₃×C₅ ⋊ #467
Sylow	C₃ ⋊ C₅×#467

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×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×C₅	D₄
hypercenter	C₅×C₂²⋊Q₈	D₃
hypocenter	C₃	C₅×#467
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×C₂²⋊Q₈

228 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 90 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)(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21);(9,10)(11,12);(7,8)(10,12)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#467, C₂×C₅×#5051, D₅×#5051, …
maximal normal subgroup of	C₅×D₃×#467, C₂×C₅×#5051, D₅×#5051, …
maximal quotient of	C₅×#71101, C₅×#71117, C₅×#71133, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4745);