C₅×#5199 #11750698

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

Aliases

direct product

C₅×#5199

Splittings

Fitting	C₃×C₅×AGL(1,ℤ/8) ⋊ C₂
Hall	C₃×C₅ ⋊ #472
Sylow	C₃ ⋊ C₅×#472

Subgroups

	subgroup	G/H
center	C₂×C₅	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×AGL(1,ℤ/8)	C₂
socle	C₂×C₃×C₅	#121
hypercenter	C₅×AGL(1,ℤ/8)	D₃
hypocenter	C₃	C₅×#472
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×AGL(1,ℤ/8)

216 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 74 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)(11,19)(12,22)(16,17)(18,23)(20,21);(9,11)(10,14)(12,19)(13,21)(15,18)(16,23)(17,24)(20,22);(7,8)(9,10)(11,18)(12,15)(13,17)(14,20)(16,22)(19,24)(21,23)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#472, C₂×C₅×#5199, D₅×#5199, …
maximal normal subgroup of	C₅×D₃×#472, C₂×C₅×#5199, D₅×#5199, …
maximal quotient of	C₅×#71005, C₅×#71021, C₅×#71133, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4893);