C₅×#69154 #14316781

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

Aliases

direct product

C₅×#69154

Splittings

Fitting	C₃×C₅×#354 ⋊ C₂
Hall	C₃×C₅ ⋊ #2199
Sylow	C₃ ⋊ C₅×#2199

Subgroups

	subgroup	G/H
center	C₂×C₅	#4741
commutator, cosocle	C₂×C₃×D₄	C₂³×C₅
Frattini	C₂×D₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#354	C₂
socle	C₂×C₃×C₅	B(4,𝔽₂)
hypercenter	C₅×#354	D₃
hypocenter	C₃	C₅×#2199
LCS	G ▷ C₂×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×D₄ ◁ C₅×#354

540 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,15)(9,17)(11,18)(14,20)(22,23);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(23,24);(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(23,24)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42479);