C₅×C₃⋊#14524208 #44570602

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

Aliases

direct product	C₅×C₃⋊#14524208
unambiguous semidirect product	C₃⋊(C₅×#14524208)

Splittings

Fitting, socle	C₃×C₅×C₁₉² ⋊ C₂×C₃
derived	C₃×C₁₉² ⋊ C₂×C₃×C₅
Hall	C₃×C₅×#11762245 ⋊ C₂
Hall	C₃×#11762245 ⋊ C₂×C₅
Hall	C₅×C₁₉² ⋊ C₃×D₃
Sylow	C₁₉² ⋊ C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#14524208
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

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 186);

C5×C3⋊#14524208 #44570602