C₅×#78579 #14418023

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#78579

Splittings

Fitting	C₃×C₅×#405 ⋊ C₂
Hall	C₃×C₅ ⋊ #2955
Sylow	C₃ ⋊ C₅×#2955

Subgroups

	subgroup	G/H
center	C₅×C₈	C₂×D₁₂
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#405	C₂
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₅×#5748	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#405	D₃
hypocenter	C₃	C₅×#2955
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₅×#405

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 143721);

C5×#78579 #14418023