C₅×#5748 #11756133

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#5748

Splittings

Fitting	C₂×C₃×C₅×M₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #575
Sylow	C₃ ⋊ C₅×#575

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₅×M₁₆	D₃
hypocenter	C₃	C₅×#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×M₁₆

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

normal subgroups only

This group has 524 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) = 240

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#69035, C₅×#68947, #14301208, …
maximal normal subgroup of	C₅×#69035, C₅×#68947, #14301208, …
maximal quotient of	C₅×#83768, C₅×#83752, C₅×#78494, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10328);