C₅×C₃⋊((C₁₉⋊C₃)⋊D₁₉) #44570598

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 855

A minimal faithful permutation representation — 2 generators on 46 points:

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

Conjugacy classes & character table

1305 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 182);

C5×C3⋊((C19⋊C3)⋊D19) #44570598