C₅×#5045 #11750544

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

Aliases

direct product

C₅×#5045

Splittings

Hall	C₃×C₅ ⋊ #485
Sylow	C₃ ⋊ C₅×#485

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×C₅	D₄
hypercenter	C₅×C₂²⋊Q₈	D₃
hypocenter	C₃	C₅×#485
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×C₂²⋊Q₈

192 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 480

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

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#485, C₂×C₅×#5045, D₅×#5045, …
maximal normal subgroup of	C₅×D₃×#485, C₂×C₅×#5045, D₅×#5045, …
maximal quotient of	C₅×#69503, C₅×#69506, C₅×#71693, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4739);