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C5×C3⋊D777 #37552145

order23310
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D777
unambiguous semidirect productC3⋊(C5×D777)
unambiguous semidirect productC7⋊(C5×C3⋊D111)
unambiguous semidirect productC37⋊(C5×C3⋊D21)

Splittings

Fitting, socle, HallC32×C5×C7×C37 ⋊ C2
derived, HallC32×C7×C37 ⋊ C2×C5
HallC32×C5×C37 ⋊ D7
HallC5×C7×C37 ⋊ C3⋊D3
HallC32×C37 ⋊ C5×D7
HallC32×C5×C7 ⋊ D37
HallC7×C37 ⋊ C5×C3⋊D3
HallC5×C37 ⋊ C3⋊D21
HallC32×C7 ⋊ C5×D37
HallC32×C5 ⋊ D259
SylowC37 ⋊ C5×C3⋊D21
HallC5×C7 ⋊ C3⋊D111
SylowC32 ⋊ C5×D259
SylowC7 ⋊ C5×C3⋊D111

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D777
commutator, cosocle, hypocenterC32×C7×C37C2×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C37C2
LCSG ▷ C32×C7×C37
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 11655

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5)(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)(33,34)(35,36)(37,38)(39,40)(41,42)(44,45)(46,47)(48,49)(51,52)(54,55);(6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8)(43,44,46,48,49,47,45)(53,54,55);(50,51,52)

Conjugacy classes & character table

5835 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 164);