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C2×C13×C3⋊(C5⋊(C7⋊D7)) #50304531

order38220
automorphism?
homologyAbelianization (H1) = C22×C13 · Schur multiplier (H2) = C2×C7

Aliases

direct productC2×C13×C3⋊(C5⋊(C7⋊D7))
unambiguous semidirect product(C2×C3)⋊(C13×C5⋊(C7⋊D7))
unambiguous semidirect product(C2×C5)⋊(C13×C3⋊(C7⋊D7))
unambiguous semidirect product(C2×C7)⋊(C13×D105)

Splittings

Fitting, socleC2×C3×C5×C72×C13 ⋊ C2
derived, HallC3×C5×C72 ⋊ C22×C13
HallC3×C5×C72×C13 ⋊ C22
HallC5×C72×C13 ⋊ C2×D3
HallC3×C72×C13 ⋊ C2×D5
HallC72×C13 ⋊ C2×D15
HallC5×C72 ⋊ C2×D3×C13
HallC3×C5×C13 ⋊ C2×C7⋊D7
HallC3×C72 ⋊ C2×D5×C13
HallC5×C13 ⋊ C2×C3⋊(C7⋊D7)
SylowC72 ⋊ C2×C13×D15
HallC3×C13 ⋊ C2×C5⋊(C7⋊D7)
HallC3×C5 ⋊ C2×C13×C7⋊D7
SylowC5 ⋊ C2×C13×C3⋊(C7⋊D7)
SylowC3 ⋊ C2×C13×C5⋊(C7⋊D7)

Subgroups

subgroupG/H
center, hypercenterC2×C13C3⋊(C5⋊(C7⋊D7))
commutator, cosocle, hypocenterC3×C5×C72C22×C13
FrattiniC1G
Fitting, socleC2×C3×C5×C72×C13C2
LCSG ▷ C3×C5×C72
UCSC1 ◁ C2×C13

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 19110

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

9594 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 1217);