C₅×#71862 #14313720

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#71862

Splittings

Hall	C₃×C₅ ⋊ #2134
Sylow	C₃ ⋊ C₅×#2134

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#834
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×Q₈	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#535	C₂
socle	C₂×C₃×C₅	C₂×#99
cosocle	C₂×C₃×Q₈	C₂³×C₅
hypercenter	C₅×#535	D₃
hypocenter	C₃	C₅×#2134
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ C₅×#535

524 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4,5)(6,7,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23);(10,15)(11,12)(16,18)(17,23)(19,20)(22,24);(7,8)(9,10,13,17,14,18,21,24)(11,15,19,22,12,16,20,23)

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, 39418);