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C5×C3⋊D741 #36372735

order22230
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D741
unambiguous semidirect productC3⋊(C5×D741)
unambiguous semidirect productC13⋊(C5×C3⋊D57)
unambiguous semidirect productC19⋊(C5×C3⋊D39)

Splittings

Fitting, socle, HallC32×C5×C13×C19 ⋊ C2
derived, HallC32×C13×C19 ⋊ C2×C5
HallC5×C13×C19 ⋊ C3⋊D3
HallC32×C5×C19 ⋊ D13
HallC32×C5×C13 ⋊ D19
HallC13×C19 ⋊ C5×C3⋊D3
HallC32×C19 ⋊ C5×D13
HallC32×C13 ⋊ C5×D19
HallC5×C19 ⋊ C3⋊D39
HallC5×C13 ⋊ C3⋊D57
HallC32×C5 ⋊ D247
SylowC19 ⋊ C5×C3⋊D39
SylowC13 ⋊ C5×C3⋊D57
SylowC32 ⋊ C5×D247

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D741
commutator, cosocle, hypocenterC32×C13×C19C2×C5
FrattiniC1G
Fitting, socleC32×C5×C13×C19C2
LCSG ▷ C32×C13×C19
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 11115

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

5565 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22230, 164);