C₅×#5242 #11750741

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

Aliases

direct product

C₅×#5242

Splittings

Hall	C₃×C₅ ⋊ #455
Sylow	C₃ ⋊ C₅×#455

Subgroups

	subgroup	G/H
center	C₂×C₅	#953
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₅	#120
hypercenter	C₅×H(5,𝔽₂)	D₃
hypocenter	C₃	C₅×#455
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×H(5,𝔽₂)

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

normal subgroups only

This group has 304 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) = 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)(12,22)(15,20)(17,24)(19,21);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(7,8)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71000, C₅×#70998, #14316692, …
maximal normal subgroup of	C₅×#71000, C₅×#70998, #14316692, …
maximal quotient of	C₅×#71846, C₅×#71893, C₅×#71885, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4936);