D₃₁×D₆₁⋊C₅ #49180671

order	37820
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃₁×D₆₁⋊C₅
unambiguous semidirect product	D₆₁⋊(C₅×D₃₁)
unambiguous semidirect product	(C₃₁×D₆₁⋊C₅)⋊C₂
unambiguous semidirect product	(D₃₁×C₆₁⋊C₅)⋊C₂
unambiguous semidirect product	((C₆₁⋊C₅)⋊D₃₁)⋊C₂

Splittings

Fitting, socle, derived, Hall	C₃₁×C₆₁ ⋊ C₂²×C₅
Hall	C₃₁×C₆₁⋊C₅ ⋊ C₂²
Hall	D₃₁×D₆₁ ⋊ C₅
Hall	C₆₁⋊C₅ ⋊ C₂×D₃₁
Sylow	C₆₁ ⋊ C₂×C₅×D₃₁
Sylow	C₃₁ ⋊ C₂×D₆₁⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₁×C₆₁	C₂²×C₅
LCS	G ▷ C₃₁×C₆₁
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 1891

A minimal faithful permutation representation — 2 generators on 92 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(33,37,38,42,59,34,35,43,45,53)(36,50,51,60,75,40,41,46,68,58)(39,62,63,64,76,47,48,67,80,81)(44,49,74,82,79,56,57,65,89,70)(52,66,87,71,72,55,69,83,54,88)(61,90,84,73,91,77,78,92,86,85);(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)(30,31)(32,33,36,42,54,73,60,75,56,35,39,49,66,84,87,78,91,63,41,52,38,46,61,74,89,88,81,53,70,48,64,80,62,79,59,76,72,82,65,77,51,43,55,50,67,85,90,83,92,69,57,47,37,44,58,68,86,71,45,40,34)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37820, 51);

D31×D61⋊C5 #49180671