C₅×C₃⋊#14524207 #44570605

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

Aliases

direct product	C₅×C₃⋊#14524207
unambiguous semidirect product	C₃⋊(C₅×#14524207)

Splittings

Fitting, socle	C₃×C₅×C₁₉² ⋊ C₂×C₃
derived	C₃×C₁₉² ⋊ C₂×C₃×C₅
Hall	C₃×C₅×#11762246 ⋊ C₂
Hall	C₃×#11762246 ⋊ C₂×C₅
Hall	C₅×C₁₉² ⋊ C₃×D₃
Sylow	C₁₉² ⋊ C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#14524207
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)(29,33,34,30,31,35)(32,40,41,36,37,45)(38,46,44,43,42,39);(6,7,8)(9,10,13,19,25,22,26,12,16,21,27,15,14,20,18,23,24,17,11)(28,29,32,37,42,45,44,34,33,38,43,31,35,39,41,46,40,36,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 189);

C5×C3⋊#14524207 #44570605