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C13×C3⋊(C5⋊D95) #49150336

order37050
automorphism?
homologyAbelianization (H1) = C2×C13 · Schur multiplier (H2) = C5

Aliases

direct productC13×C3⋊(C5⋊D95)
unambiguous semidirect productC3⋊(C13×C5⋊D95)
unambiguous semidirect productC5⋊(C13×D285)
unambiguous semidirect productC19⋊(C13×C3⋊(C5⋊D5))

Splittings

Fitting, socle, HallC3×C52×C13×C19 ⋊ C2
derived, HallC3×C52×C19 ⋊ C2×C13
HallC52×C13×C19 ⋊ D3
HallC3×C52×C13 ⋊ D19
HallC3×C13×C19 ⋊ C5⋊D5
HallC52×C19 ⋊ D3×C13
HallC52×C13 ⋊ D57
HallC13×C19 ⋊ C3⋊(C5⋊D5)
HallC3×C52 ⋊ C13×D19
HallC3×C19 ⋊ C13×C5⋊D5
HallC3×C13 ⋊ C5⋊D95
SylowC52 ⋊ C13×D57
SylowC19 ⋊ C13×C3⋊(C5⋊D5)
SylowC3 ⋊ C13×C5⋊D95

Subgroups

subgroupG/H
center, hypercenterC13C3⋊(C5⋊D95)
commutator, cosocle, hypocenterC3×C52×C19C2×C13
FrattiniC1G
Fitting, socleC3×C52×C13×C19C2
LCSG ▷ C3×C52×C19
UCSC1 ◁ C13

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 18525

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

9282 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37050, 162);