C₅×#71490 #14316508

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

Aliases

direct product

C₅×#71490

Splittings

Fitting	C₂×C₃×C₅×SD₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #2194
Sylow	C₃ ⋊ C₅×#2194

Subgroups

	subgroup	G/H
center	C₂×C₅	#5156
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×SD₃₂	C₂
socle	C₂×C₃×C₅	#447
prime radical	C₅×#5757	C₂
hypercenter	C₂×C₅×SD₃₂	D₃
hypocenter	C₃	C₅×#2194
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×SD₃₂

432 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 94 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)(6,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,17)(18,35)(19,36)(20,30)(22,27)(25,33)(26,28)(29,31)(32,40)(34,38)(37,39);(7,8)(9,11)(10,15)(12,26)(13,22)(14,24)(16,17)(18,32)(19,34)(20,25)(21,28)(23,27)(29,39)(30,33)(31,37)(35,40)(36,38);(9,10,14,19)(11,20,24,35)(12,25,29,40)(13,30,27,18)(15,31,34,21)(16,23,36,22)(17,37,38,28)(26,32,39,33)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42206);