C₅×#15691772⋊C₂ #44540712

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

Aliases

direct product	C₅×#15691772⋊C₂
unambiguous semidirect product	#15691772⋊(C₂×C₅)
unambiguous semidirect product	((C₂₃⋊D₂₃)⋊C₃)⋊(C₂×C₅)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#15691772⋊C₂
commutator, cosocle, hypocenter	C₂₃²⋊C₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₂₃²	C₂×D₃
LCS	G ▷ C₂₃²⋊C₃
UCS	C₁ ◁ C₅

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 345

A minimal faithful permutation representation — 2 generators on 74 points:

(1,2,3,4,5)(7,51)(9,40)(10,22)(11,52)(13,18)(15,31)(16,37)(17,27)(19,38)(20,43)(21,28)(23,64)(24,30)(29,44)(33,39)(34,57)(35,60)(36,63)(42,61)(46,56)(47,59)(49,58)(54,55);(6,7,9)(8,24,15,12,13,27)(10,28,25,17,18,14)(11,29,41,21,22,48)(16,50,31,30,32,19)(20,61,62,37,38,72)(23,64,65,42,43,74)(26,34,46,45,40,51)(33,68,44,52,53,55)(35,47,70,56,57,66)(36,49,71,59,60,67)(39,54,73,58,63,69)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 50);

C5×#15691772⋊C2 #44540712