C₅×C₉⋊(C₁₉⋊D₁₉) #44570488

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁₉

Aliases

direct product	C₅×C₉⋊(C₁₉⋊D₁₉)
unambiguous semidirect product	C₉⋊(C₅×C₁₉⋊D₁₉)
unambiguous semidirect product	C₁₉⋊(C₅×D₁₇₁)

Splittings

Fitting, Hall	C₅×C₉×C₁₉² ⋊ C₂
derived, Hall	C₉×C₁₉² ⋊ C₂×C₅
Hall	C₅×C₁₉² ⋊ D₉
Sylow	C₁₉² ⋊ C₅×D₉
Hall	C₅×C₉ ⋊ C₁₉⋊D₁₉
Sylow	C₉ ⋊ C₅×C₁₉⋊D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₉⋊(C₁₉⋊D₁₉)
commutator, cosocle, hypocenter	C₉×C₁₉²	C₂×C₅
Frattini	C₃	C₅×C₃⋊(C₁₉⋊D₁₉)
Fitting	C₅×C₉×C₁₉²	C₂
socle	C₃×C₅×C₁₉²	D₃
LCS	G ▷ C₉×C₁₉²
UCS	C₁ ◁ C₅

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)(7,9)(8,12)(10,13)(11,14)(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);(6,11,9,8,13,10,12,7,14)(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

8130 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 72);

C5×C9⋊(C19⋊D19) #44570488