C₅×C₃⋊#14523747 #44540740

order	31740
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₂₃

Aliases

direct product	C₅×C₃⋊#14523747
unambiguous semidirect product	C₃⋊(C₅×#14523747)

Splittings

Fitting, socle, Hall	C₃×C₅×C₂₃² ⋊ C₄
derived, Hall	C₃×C₂₃² ⋊ C₄×C₅
Hall	C₅×C₂₃² ⋊ Dic₃
Sylow	C₂₃² ⋊ C₅×Dic₃
Hall	C₃×C₅ ⋊ #14523747
Sylow	C₃ ⋊ C₅×#14523747

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#14523747
commutator, hypocenter	C₃×C₂₃²	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₂₃²	C₄
prime radical	C₃×C₅×C₂₃⋊D₂₃	C₂
cosocle	C₃×C₂₃⋊D₂₃	C₂×C₅
LCS	G ▷ C₃×C₂₃²
UCS	C₁ ◁ C₅

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 690

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,4,5)(7,8)(9,10)(11,14,15,24)(12,17,18,30)(13,21,22,36)(16,27,28,38)(19,32,25,37)(20,33,34,31)(23,39,40,51)(26,43,41,49)(29,35,45,48)(42,52,50,47)(44,46,53,54);(6,7,8)(9,11,16,29,46,43,21,37,17,31,47,51,39,52,33,30,32,36,49,54,45,28,15)(10,38,44,13,12,50,23,20,19,41,35,14,24,48,26,25,34,40,42,18,22,53,27)

Conjugacy classes & character table

2010 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 78);

C5×C3⋊#14523747 #44540740