C₅×#5062 #11750561

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

Aliases

direct product

C₅×#5062

Splittings

Fitting	C₃×C₅×#124 ⋊ C₂
Hall	C₃×C₅ ⋊ #454
Sylow	C₃ ⋊ C₅×#454

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
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#454
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#124

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

normal subgroups only

This group has 292 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)(10,15)(12,21)(14,22)(16,17)(18,20)(23,24);(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,23,16,22);(7,8)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#67542, C₅×#67460, #14279355, …
maximal normal subgroup of	C₅×#67542, C₅×#67460, #14279355, …
maximal quotient of	C₅×#69702, C₅×#69703, C₅×#69704, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4756);