C₅×#4754 #11750253

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

Aliases

direct product

C₅×#4754

Splittings

Fitting	C₃×C₅×#101 ⋊ C₂
Hall	C₃×C₅ ⋊ #471
Sylow	C₃ ⋊ C₅×#471

Subgroups

	subgroup	G/H
center	C₂×C₅	#884
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#101	C₂
socle	C₂×C₃×C₅	#121
prime radical	C₅×#908	C₂
hypercenter	C₅×#101	D₃
hypocenter	C₃	C₅×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#101

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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)(9,12,14,21)(10,15,17,18)(11,16,19,24)(13,22,23,20);(9,11,13,18,14,19,23,15)(10,12,16,20,17,21,24,22);(7,8)(9,10)(11,12)(13,16)(14,17)(15,22)(18,20)(19,21)(23,24)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#471, C₂×C₅×#4754, D₅×#4754, …
maximal normal subgroup of	C₅×D₃×#471, C₂×C₅×#4754, D₅×#4754, …
maximal quotient of	C₅×#68440, C₅×#68464, C₅×#68442, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4448);