C₅×#67825 #14279648

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

Aliases

direct product

C₅×#67825

Splittings

Hall	C₃×C₅ ⋊ #1419
Sylow	C₃ ⋊ C₅×#1419

Subgroups

	subgroup	G/H
center	C₂×C₅	#4562
commutator	C₃×C₁₆	C₂×C₄×C₅
Frattini	C₂×C₁₆	C₂×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×D₁₆	C₂
socle	C₂×C₃×C₅	#357
cosocle	C₂×C₃×C₁₆	C₂²×C₅
hypercenter	C₂×C₅×D₁₆	D₃
hypocenter	C₃	C₅×#1419
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₂×C₅×D₁₆

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 40 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 5346);