C₅×#4914 #11750413

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

Aliases

direct product

C₅×#4914

Splittings

Fitting	C₃×C₅×SD₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #509
Sylow	C₃ ⋊ C₅×#509

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₅
Frattini	C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×SD₃₂	C₂
socle	C₂×C₃×C₅	C₂×D₈
hypercenter	C₅×SD₃₂	D₃
hypocenter	C₃	C₅×#509
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×SD₃₂

180 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 62 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)(9,10,13,14)(11,18,20,24)(12,21,19,16)(15,23,22,17);(10,15)(11,17)(12,19)(14,22)(16,24)(18,21)(20,23);(7,8)(10,14)(15,22)(16,21)(18,24)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#509, C₂×C₅×#4914, D₅×#4914, …
maximal normal subgroup of	C₅×D₃×#509, C₂×C₅×#4914, D₅×#4914, …
maximal quotient of	C₅×#69221, C₅×#69190, C₅×#69346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4608);