C₅×#71709 #14283414

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

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₅×#71709

Splittings

Hall	C₃×C₅ ⋊ #1513
Sylow	C₃ ⋊ C₅×#1513

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#525	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#525	D₃
hypocenter	C₃	C₅×#1513
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₅×#525

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 24 points:

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

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, 9112);

C5×#71709 #14283414