C₉×C₅⋊(C₁₉⋊D₁₉) #44570491

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁₉

Aliases

direct product	C₉×C₅⋊(C₁₉⋊D₁₉)
unambiguous semidirect product	C₅⋊(C₉×C₁₉⋊D₁₉)

Splittings

Fitting, Hall	C₅×C₉×C₁₉² ⋊ C₂
derived, Hall	C₅×C₁₉² ⋊ C₂×C₉
Hall	C₉×C₁₉² ⋊ D₅
Sylow	C₁₉² ⋊ C₉×D₅
Hall	C₅×C₉ ⋊ C₁₉⋊D₁₉
Sylow	C₅ ⋊ C₉×C₁₉⋊D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	C₅⋊(C₁₉⋊D₁₉)
commutator, hypocenter	C₅×C₁₉²	C₂×C₉
Frattini	C₃	C₃×C₅⋊(C₁₉⋊D₁₉)
Fitting	C₅×C₉×C₁₉²	C₂
socle, cosocle	C₃×C₅×C₁₉²	C₂×C₃
prime radical	C₃×C₅⋊(C₁₉⋊D₁₉)	C₃
LCS	G ▷ C₅×C₁₉²
UCS	C₁ ◁ C₉

264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 16245

A minimal faithful permutation representation — 3 generators on 52 points:

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

Conjugacy classes & character table

8136 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 75);

C9×C5⋊(C19⋊D19) #44570491