C₅×#71088 #14318514

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#71088

Splittings

Hall	C₃×C₅ ⋊ #2238
Sylow	C₃ ⋊ C₅×#2238

Subgroups

	subgroup	G/H
center	C₂×C₅	#5034
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#469	C₂
socle	C₂×C₃×C₅	#481
hypercenter	C₅×#469	D₃
hypocenter	C₃	C₅×#2238
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#469

268 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 90 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)(7,8)(9,10,12,21,13,15,20,16)(11,19,17,22,18,23,24,14);(6,7,8)(11,18)(14,22)(17,24)(19,23);(10,14)(11,17)(12,20)(15,22)(16,19)(18,24)(21,23)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 44212);