C₅×#5688 #11756073

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#5688

Splittings

Fitting	C₃×C₅×#125 ⋊ C₂
Hall	C₃×C₅ ⋊ #533
Sylow	C₃ ⋊ C₅×#533

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#125	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₃	C₂³
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#533
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#125

444 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 214 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 — 4 generators on 24 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#68362, #14284022, C₅×#68365, …
maximal normal subgroup of	C₅×#68362, #14284022, C₅×#68365, …
maximal quotient of	C₅×#74148, C₅×#74938, C₅×#75963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10268);