C₅×#4847 #11750346

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#4847

Splittings

Fitting	C₃×C₅×C₄⋊C₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #478
Sylow	C₃ ⋊ C₅×#478

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×D₁₂	C₂²
hypercenter	C₅×C₄⋊C₈	D₃
hypocenter	C₃	C₅×#478
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₄⋊C₈

204 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 480

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(6,7,8)(10,14)(11,15)(17,18,23,24,20,22,19,21);(9,11,12,15)(10,13,14,16)(18,22)(21,24);(7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(19,23)(22,24)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#478, C₂×C₅×#4847, D₅×#4847, …
maximal normal subgroup of	C₅×D₃×#478, C₂×C₅×#4847, D₅×#4847, …
maximal quotient of	C₅×#70480, C₅×#68670, C₅×#68671, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4541);