C₅×#68205 #14289823

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

Aliases

direct product

C₅×#68205

Splittings

Hall	C₃×C₅ ⋊ #1638
Sylow	C₃ ⋊ C₅×#1638

Subgroups

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×D₄⋊C₈	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₃×C₅×D₄	D₄
hypercenter	C₅×D₄⋊C₈	D₃
hypocenter	C₃	C₅×#1638
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×D₄⋊C₈

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 960

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

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

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, 15521);

C5×#68205 #14289823