C₅×#5147 #11750646

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

Aliases

direct product

C₅×#5147

Splittings

Fitting	C₂×C₃×C₅×D₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #509
Sylow	C₃ ⋊ C₅×#509

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#213
commutator, cosocle	C₃×C₈	C₂³×C₅
Frattini	C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×D₈	C₂
socle	C₂×C₃×C₅	C₂×D₈
hypercenter	C₂×C₅×D₈	D₃
hypocenter	C₃	C₅×#509
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×D₈

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

normal subgroups only

This group has 180 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)(10,16)(11,13)(12,20)(15,17)(18,24)(19,21)(22,23);(9,11,14,13)(10,15,18,22)(12,19,21,20)(16,23,24,17);(7,8)(9,10)(11,15)(12,16)(13,22)(14,18)(17,20)(19,23)(21,24)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#70918, C₅×#69254, #14315886, …
maximal normal subgroup of	C₅×#70918, C₅×#69254, #14315886, …
maximal quotient of	C₅×#70598, C₅×#72033, C₅×#70596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4841);