C₅×#5239 #11750738

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

Aliases

direct product

C₅×#5239

Splittings

Hall	C₃×C₅ ⋊ #452
Sylow	C₃ ⋊ C₅×#452

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₂×C₅×#41	D₃
hypocenter	C₃	C₅×#452
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 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)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22);(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(7,8)(9,10)(11,12)(13,14,18,21)(15,22,24,27)(16,20,17,25)(19,26,28,23)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#452, C₂×C₅×#5239, D₅×#5239, …
maximal normal subgroup of	C₅×D₃×#452, C₂×C₅×#5239, D₅×#5239, …
maximal quotient of	C₅×#71889, C₅×#71885, C₅×#71887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4933);