C₅×#68061 #14293600

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#68061

Splittings

Fitting	C₃×C₅×C₈² ⋊ C₂
Hall	C₃×C₅ ⋊ #1711
Sylow	C₃ ⋊ C₅×#1711

Subgroups

	subgroup	G/H
center	C₂²×C₅	#874
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	#127
prime radical	C₅×#874	C₂²
hypercenter	C₅×C₈²	D₃
hypocenter	C₃	C₅×#1711
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈²

356 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 960

A minimal faithful permutation representation — 3 generators on 40 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 19298);