C₅×#5836 #11756221

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

Aliases

direct product

C₅×#5836

Splittings

Fitting	C₃×C₅×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₅ ⋊ #577
Sylow	C₃ ⋊ C₅×#577

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₅×H(5,𝔽₂)	D₃
hypocenter	C₃	C₅×#577
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41 ◁ C₅×H(5,𝔽₂)

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

normal subgroups only

This group has 516 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)(9,13,15,12)(10,18,19,24)(11,20,14,21)(16,17,23,22);(9,12,15,13)(10,17,19,22)(11,20,14,21)(16,18,23,24);(9,11,15,14)(10,16,19,23)(12,21,13,20)(17,24,22,18);(7,8)(9,10)(11,16)(12,17)(13,22)(14,23)(15,19)(18,20)(21,24)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#80280, C₅×#84009, #14457718, …
maximal normal subgroup of	C₅×#80280, C₅×#84009, #14457718, …
maximal quotient of	C₅×#84099, C₅×#82570, C₅×#82561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10416);