C₅×C₃⋊#14524212 #44570595

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

Aliases

direct product	C₅×C₃⋊#14524212
unambiguous semidirect product	C₃⋊(C₅×#14524212)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#14524212
commutator, cosocle, hypocenter	C₃×#11762246	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₉²	D₃
LCS	G ▷ C₃×#11762246
UCS	C₁ ◁ C₅

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 570

A minimal faithful permutation representation — 3 generators on 46 points:

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

Conjugacy classes & character table

1065 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 179);

C5×C3⋊#14524212 #44570595