C₅×#71716 #14313137

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

Aliases

direct product

C₅×#71716

Splittings

Hall	C₃×C₅ ⋊ #2120
Sylow	C₃ ⋊ C₅×#2120

Subgroups

	subgroup	G/H
center	C₄×C₅	#834
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂×D₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#525	C₂
socle	C₂×C₃×C₅	C₂×#99
cosocle	C₂×C₃×D₄	C₂³×C₅
hypercenter	C₅×#525	D₃
hypocenter	C₃	C₅×#2120
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ C₅×#525

528 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)(6,7,8)(9,11,13,18)(10,15,16,22)(12,17,20,24)(14,21,19,23);(10,14)(15,21)(16,19)(22,23);(7,8)(9,10)(11,15)(12,19,20,14)(13,16)(17,23,24,21)(18,22)

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, 38835);

C5×#71716 #14313137