C₅×C₃⋊(C₃⋊(C₁₉⋊D₁₉)) #44570634

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₃×C₁₉

Aliases

direct product	C₅×C₃⋊(C₃⋊(C₁₉⋊D₁₉))
unambiguous semidirect product	C₃⋊(C₅×C₃⋊(C₁₉⋊D₁₉))
unambiguous semidirect product	C₁₉⋊(C₅×C₃⋊D₅₇)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₉² ⋊ C₂
derived, Hall	C₃²×C₁₉² ⋊ C₂×C₅
Hall	C₅×C₁₉² ⋊ C₃⋊D₃
Sylow	C₁₉² ⋊ C₅×C₃⋊D₃
Hall	C₃²×C₅ ⋊ C₁₉⋊D₁₉
Sylow	C₃² ⋊ C₅×C₁₉⋊D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(C₃⋊(C₁₉⋊D₁₉))
commutator, cosocle, hypocenter	C₃²×C₁₉²	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₉²	C₂
LCS	G ▷ C₃²×C₁₉²
UCS	C₁ ◁ C₅

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:

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

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, 218);

C5×C3⋊(C3⋊(C19⋊D19)) #44570634