C₃×C₅×D₅₃ #13097548

order	1590
isoclinism	isoclinic class of D₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×D₅₃
unambiguous semidirect product	C₅₃⋊(C₂×C₃×C₅)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₅₃
commutator, cosocle, hypocenter	C₅₃	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₅₃	C₂
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₃×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);(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

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

Relationships

maximal subgroup of	C₃×C₅×#6157, C₃×#11761508, C₃×D₅×D₅₃, …
maximal normal subgroup of	C₃×C₅×#6157, C₃×#11761508, C₃×D₅×D₅₃, …
maximal quotient of	C₃×C₅×Dic₅₃, C₂×C₃×C₅×D₅₃, C₅×C₉×D₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1590, 1);

C3×C5×D53 #13097548