C₅×#5683 #11756068

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₅×#5683

Splittings

Fitting	C₃×C₅×#124 ⋊ C₂
Hall	C₃×C₅ ⋊ H(3,𝔽₄)
Sylow	C₃ ⋊ C₅×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×H(3,𝔽₄)
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#124

520 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 182 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)(9,11,12,16)(10,14,13,15)(18,22)(19,23);(10,13)(11,16)(17,19)(18,21)(20,23)(22,24);(11,16)(14,15)(17,18)(19,24)(20,22)(21,23);(7,8)(9,10)(11,15)(12,13)(14,16)(19,23)(21,24)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71326, C₅×#68350, #14290891, …
maximal normal subgroup of	C₅×#71326, C₅×#68350, #14290891, …
maximal quotient of	C₅×#76434, C₅×#75840, C₅×#75493, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10263);