C₅×#68699 #14298763

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#68699

Splittings

Hall	C₃×C₅ ⋊ #1833
Sylow	C₃ ⋊ C₅×#1833

Subgroups

	subgroup	G/H
center	C₄×C₅	#872
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×H(3,ℤ/4)	C₂
socle	C₂×C₃×C₅	#382
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×H(3,ℤ/4)	D₃
hypocenter	C₃	C₅×#1833
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#117 ◁ C₅×H(3,ℤ/4)

344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(6,7,8)(9,12)(10,16,17,29)(11,20)(13,24)(14,25)(15,27,19,31)(18,23,30,38)(21,34)(22,35)(26,39)(28,37,32,33)(36,40);(7,8)(9,11,13,21,14,22,26,36)(10,15,18,28,17,19,30,32)(12,20,24,34,25,35,39,40)(16,27,23,37,29,31,38,33);(9,10)(11,19,22,15)(12,23,25,38)(13,18)(14,17)(16,24,29,39)(20,33)(21,32,36,28)(26,30)(27,34)(31,40)(35,37)

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, 24461);

C5×#68699 #14298763