C₅×#5061 #11750560

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

Aliases

direct product

C₅×#5061

Splittings

Fitting	C₃×C₅×#124 ⋊ C₂
Hall	C₃×C₅ ⋊ #455
Sylow	C₃ ⋊ C₅×#455

Subgroups

	subgroup	G/H
center	C₂×C₅	#937
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#124	C₂
socle	C₂×C₃×C₅	#120
prime radical	C₅×#1004	C₂
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#455
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#124

304 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,11,13,19)(10,14,16,23)(12,22,21,18)(15,24,20,17);(12,21)(14,23)(15,20)(18,22);(7,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,21)(19,24)(22,23)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#67546, C₅×#67461, #14279332, …
maximal normal subgroup of	C₅×#67546, C₅×#67461, #14279332, …
maximal quotient of	C₅×#69699, C₅×#69701, C₅×#69700, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4755);