C₅×#84110 #14425704

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#84110

Splittings

Fitting	C₂×C₃×C₅×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₅ ⋊ #3026
Sylow	C₃ ⋊ C₅×#3026

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#953
commutator, cosocle	C₂³×C₃	C₂⁴×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₅	C₂×#120
hypercenter	C₂×C₅×H(5,𝔽₂)	D₃
hypocenter	C₃	C₅×#3026
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂⁴×C₅ ◁ C₂×C₅×H(5,𝔽₂)

1600 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 322 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 — 4 generators on 24 points:

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

Conjugacy classes & character table

315 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 151402);