C₅×#69115 #14314920

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#69115

Splittings

Hall	C₃×C₅ ⋊ #2166
Sylow	C₃ ⋊ C₅×#2166

Subgroups

	subgroup	G/H
center	C₄×C₅	#877
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#349	C₂
socle	C₂×C₃×C₅	#432
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#349	D₃
hypocenter	C₃	C₅×#2166
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#349

244 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 40 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 40618);