C₅×C₃⋊(C₃⋊#10621729) #44478300

order	30420
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₃×C₁₃

Aliases

direct product	C₅×C₃⋊(C₃⋊#10621729)
unambiguous semidirect product	C₃⋊(C₅×C₃⋊#10621729)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₃² ⋊ C₄
derived, Hall	C₃²×C₁₃² ⋊ C₄×C₅
Hall	C₅×C₁₃² ⋊ C₃⋊Dic₃
Sylow	C₁₃² ⋊ C₅×C₃⋊Dic₃
Hall	C₃²×C₅ ⋊ #10621729
Sylow	C₃² ⋊ C₅×#10621729

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(C₃⋊#10621729)
commutator, hypocenter	C₃²×C₁₃²	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃²	C₄
prime radical	C₃²×C₅×C₁₃⋊D₁₃	C₂
cosocle	C₃²×C₁₃⋊D₁₃	C₂×C₅
LCS	G ▷ C₃²×C₁₃²
UCS	C₁ ◁ C₅

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 1170

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

1950 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 889);

C5×C3⋊(C3⋊#10621729) #44478300