C₅×#84124 #14425778

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#84124

Splittings

Hall	C₃×C₅ ⋊ #3028
Sylow	C₃ ⋊ C₅×#3028

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#953
commutator, cosocle	C₂³×C₃	C₂⁴×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting, prime radical	C₃×C₅×#585	C₂
socle	C₂×C₃×C₅	C₂×#120
hypercenter	C₅×#585	D₃
hypocenter	C₃	C₅×#3028
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#585

1432 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,4,5)(6,7,8)(9,13,14,25)(10,17,18,24)(11,20,21,35)(12,22,23,37)(15,27,28,34)(16,29,30,36)(19,32,33,40)(26,38,31,39);(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,29)(18,30)(20,32)(21,33)(24,36)(25,37)(27,38)(28,31)(34,39)(35,40);(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,32)(23,33)(24,34)(25,35)(29,38)(30,31)(36,39)(37,40);(7,8)(9,10,14,18)(11,15,21,28)(12,16,23,30)(13,24,25,17)(19,31,33,26)(20,34,35,27)(22,36,37,29)(32,38,40,39)

Conjugacy classes & character table

315 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 151476);