C₅×#5022 #11750521

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#5022

Splittings

Hall	C₃×C₅ ⋊ #479
Sylow	C₃ ⋊ C₅×#479

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₄×C₅×Q₈	D₃
hypocenter	C₃	C₅×#479
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×Q₈

180 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5)(6,7,8)(17,19,21,27)(18,23,22,31)(20,29,30,26)(24,32,28,25);(9,11,12,16)(10,13,14,15)(18,22)(23,31)(24,28)(25,32);(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,25,27,32)(20,28,30,24)(23,29,31,26)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#479, C₂×C₅×#5022, D₅×#5022, …
maximal normal subgroup of	C₅×D₃×#479, C₂×C₅×#5022, D₅×#5022, …
maximal quotient of	C₅×#71627, C₅×#71628, C₅×#71629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4716);