C₅×D₆₉⋊D₂₃ #44540743

order	31740
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₆₉⋊D₂₃
unambiguous semidirect product	D₆₉⋊(C₅×D₂₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₂₃² ⋊ C₂²
derived, Hall	C₃×C₂₃² ⋊ C₂²×C₅
Hall	C₅×C₂₃² ⋊ C₂×D₃
Sylow	C₂₃² ⋊ C₂×C₅×D₃
Hall	C₃×C₅ ⋊ D₂₃²
Sylow	C₃ ⋊ C₅×D₂₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₆₉⋊D₂₃
commutator, cosocle, hypocenter	C₃×C₂₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₂₃²	C₂²
LCS	G ▷ C₃×C₂₃²
UCS	C₁ ◁ C₅

168 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)(6,7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54);(7,8)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33,35,37,39,41,43,45,47,49,51,53,54,52,50,48,46,44,42,40,38,36,34)

Conjugacy classes & character table

2175 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 81);

C5×D69⋊D23 #44540743