C₅×D₁₅₉ #13097552

order	1590
isoclinism	isoclinic class of D₁₅₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₅₉
unambiguous semidirect product	C₃⋊(C₅×D₅₃)
unambiguous semidirect product	C₅₃⋊(C₅×D₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₅₃ ⋊ C₂
derived, Hall	C₃×C₅₃ ⋊ C₂×C₅
Hall	C₅×C₅₃ ⋊ D₃
Sylow	C₅₃ ⋊ C₅×D₃
Hall	C₃×C₅ ⋊ D₅₃
Sylow	C₃ ⋊ C₅×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₅₉
commutator, cosocle, hypocenter	C₃×C₅₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₅₃	C₂
LCS	G ▷ C₃×C₅₃
UCS	C₁ ◁ C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 795

A minimal faithful permutation representation — 2 generators on 61 points:

(1,2,3,4,5)(6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(7,8)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×D₃×D₅₃, D₅×D₁₅₉, D₁₅⋊D₅₃, …
maximal normal subgroup of	C₅×D₃×D₅₃, D₅×D₁₅₉, D₁₅⋊D₅₃, …
maximal quotient of	C₅×Dic₁₅₉, C₂×C₅×D₁₅₉, C₅×D₄₇₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1590, 5);

C5×D159 #13097552