C₅×#68737 #14298581

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

Aliases

direct product

C₅×#68737

Splittings

Fitting	C₃×C₅×#339 ⋊ C₂
Hall	C₃×C₅ ⋊ #1827
Sylow	C₃ ⋊ C₅×#1827

Subgroups

	subgroup	G/H
center	C₄×C₅	#892
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#339	C₂
socle	C₂×C₃×C₅	#381
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#339	D₃
hypocenter	C₃	C₅×#1827
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#339

524 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)(10,15,16,22)(12,19)(13,20)(14,17,18,23)(21,24);(7,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,24)(22,23);(9,10)(11,18,20,14)(12,17)(13,16)(15,19,22,24)(21,23)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 24279);

C5×#68737 #14298581