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C2×C5×(C13⋊C3)⋊(C7⋊D7) #50304270

order38220
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2×C7

Aliases

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

Splittings

Fitting, socleC2×C5×C72×C13 ⋊ C2×C3
derived, HallC72×C13 ⋊ C22×C3×C5
HallC2×C5×C7⋊D91 ⋊ C3
HallC5×C72×C13⋊C3 ⋊ C22
HallC5×C72×C13 ⋊ C22×C3
HallC2×C7⋊D91 ⋊ C3×C5
HallC72×C13⋊C3 ⋊ C22×C5
HallC5×C72 ⋊ C2×D13⋊C3
HallC5×C13⋊C3 ⋊ C2×C7⋊D7
HallC5×C13 ⋊ C2×C3×C7⋊D7
SylowC72 ⋊ C2×C5×D13⋊C3
HallC13⋊C3 ⋊ C2×C5×C7⋊D7
SylowC13 ⋊ C2×C3×C5×C7⋊D7

Subgroups

subgroupG/H
center, hypercenterC2×C5(C13⋊C3)⋊(C7⋊D7)
commutator, cosocle, hypocenterC72×C13C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C72×C13C2×C3
LCSG ▷ C72×C13
UCSC1 ◁ C2×C5

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 6370

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

1760 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 956);