C₅×#68779 #14309027

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

Aliases

direct product

C₅×#68779

Splittings

Fitting	C₃×C₅×#339 ⋊ C₂
Hall	C₃×C₅ ⋊ #2031
Sylow	C₃ ⋊ C₅×#2031

Subgroups

	subgroup	G/H
center	C₄×C₅	#885
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#339	C₂
socle	C₂×C₃×C₅	#393
prime radical	C₂×C₅×#235	C₂²
hypercenter	C₅×#339	D₃
hypocenter	C₃	C₅×#2031
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#339

488 conjugacy classes of subgroups.

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)(9,11)(10,15)(12,19)(13,21)(14,20)(16,24)(17,23)(18,22);(11,19,20,21)(15,18,23,24);(7,8)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,21,23,19)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 34725);

C5×#68779 #14309027