C₃²×C₅⋊(C₁₉⋊D₁₉) #44570625

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃×C₁₉

Aliases

direct product	C₃²×C₅⋊(C₁₉⋊D₁₉)
unambiguous semidirect product	C₅⋊(C₃²×C₁₉⋊D₁₉)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₉² ⋊ C₂
derived, Hall	C₅×C₁₉² ⋊ C₂×C₃²
Hall	C₃²×C₁₉² ⋊ D₅
Sylow	C₁₉² ⋊ C₃²×D₅
Hall	C₃²×C₅ ⋊ C₁₉⋊D₁₉
Sylow	C₅ ⋊ C₃²×C₁₉⋊D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	C₅⋊(C₁₉⋊D₁₉)
commutator, cosocle, hypocenter	C₅×C₁₉²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₉²	C₂
LCS	G ▷ C₅×C₁₉²
UCS	C₁ ◁ C₃²

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 16245

A minimal faithful permutation representation — 3 generators on 49 points:

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

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, 209);

C32×C5⋊(C19⋊D19) #44570625