Finite Group Databasebeta

← Prev (#37552120) Next (#37552122) →

C3⋊(C5⋊((C37⋊C3)⋊D7)) #37552121

order23310
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C5⋊((C37⋊C3)⋊D7))
unambiguous semidirect productC5⋊(C3⋊((C37⋊C3)⋊D7))
unambiguous semidirect product(C37⋊C3)⋊D105

Splittings

Fitting, socle, derivedC3×C5×C7×C37 ⋊ C2×C3
HallC3×C5×C7×C37⋊C3 ⋊ C2
HallC3×C7×C37⋊C3 ⋊ D5
HallC3×C5×C37⋊C3 ⋊ D7
HallC5×C7×C37 ⋊ C3×D3
HallC3×C37⋊C3 ⋊ D35
HallC7×C37 ⋊ C3×D15
HallC5×C37 ⋊ C3×D21
SylowC37 ⋊ C3×D105
HallC5×C7 ⋊ C3⋊(D37⋊C3)
SylowC7 ⋊ C3⋊(C5⋊(D37⋊C3))
SylowC5 ⋊ C3⋊((C37⋊C3)⋊D7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C37C2×C3
LCSG ▷ C3×C5×C7×C37
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3885

Polycyclic presentation:

2,3^2,5,7,37|c^a=c^2,d^a=d^4,e^a=e^6,f^a=f^36,f^b=f^26

Conjugacy classes & character table

792 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 140);