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C2×C5×D1311 #39910810

order26220
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×D1311
unambiguous semidirect product(C2×C3)⋊(C5×D437)
unambiguous semidirect product(C2×C19)⋊(C5×D69)
unambiguous semidirect product(C2×C23)⋊(C5×D57)

Splittings

Fitting, socleC2×C3×C5×C19×C23 ⋊ C2
derived, HallC3×C19×C23 ⋊ C22×C5
HallC3×C5×C19×C23 ⋊ C22
HallC5×C19×C23 ⋊ C2×D3
HallC19×C23 ⋊ C2×C5×D3
HallC3×C5×C23 ⋊ C2×D19
HallC3×C5×C19 ⋊ C2×D23
HallC5×C23 ⋊ C2×D57
HallC5×C19 ⋊ C2×D69
HallC3×C23 ⋊ C2×C5×D19
HallC3×C19 ⋊ C2×C5×D23
SylowC23 ⋊ C2×C5×D57
SylowC19 ⋊ C2×C5×D69
HallC3×C5 ⋊ C2×D437
SylowC3 ⋊ C2×C5×D437

Subgroups

subgroupG/H
center, hypercenterC2×C5D1311
commutator, cosocle, hypocenterC3×C19×C23C22×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C19×C23C2
LCSG ▷ C3×C19×C23
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 52
transitive degreeμt(G) = 13110

A minimal faithful permutation representation — 2 generators on 52 points:

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

Conjugacy classes & character table

6570 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26220, 100);