C₅×#78689 #14412009

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#78689

Splittings

Fitting	C₃×C₅×#410 ⋊ C₂
Hall	C₃×C₅ ⋊ #2884
Sylow	C₃ ⋊ C₅×#2884

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1010	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#410	D₃
hypocenter	C₃	C₅×#2884
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#410

1432 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)(9,13,14,26)(10,17,18,32)(11,21,22,38)(12,23,24,25)(15,28,19,34)(16,29,30,31)(20,35,36,37)(27,39,33,40);(7,8)(9,12,14,24)(10,16,18,30)(11,20,22,36)(13,25,26,23)(15,27,19,33)(17,31,32,29)(21,37,38,35)(28,40,34,39);(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,28)(18,19)(23,35)(24,36)(25,37)(26,38)(29,39)(30,33)(31,40)(32,34);(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,33,36,27)(21,34,38,28)(23,29)(24,30)(25,31)(26,32)(35,40,37,39)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 137707);

C5×#78689 #14412009