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(#14559097⋊C3)⋊C5 #49137345

order36300
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(#14559097⋊C3)⋊C5
unambiguous semidirect product(#14559097⋊C5)⋊C3

Splittings

Fitting, socle, derivedC5×C112 ⋊ C3×C4×C5
Hall#14559097⋊C5 ⋊ C3
HallC5×(C112⋊C3)⋊C5 ⋊ C4
HallC5×#10597243 ⋊ C3×C4
HallC112⋊C3 ⋊ C5×AGL(1,𝔽5)
SylowC112 ⋊ C3×C5×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C112C3×C4×C5
prime radicalC5⋊(((C11⋊D11)⋊C3)⋊C5)C2
cosocleC5⋊(C11⋊D11)C2×C3×C5
LCSG ▷ C5×C112
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 330

Polycyclic presentation:

2,3,5,2,5,11^2|a^2=d,e^a=e^2,f^a=f^9g^3,g^a=f^2g^2,f^b=g^2,g^b=f^5g^10,f^c=f^3,g^c=g^3,e^d=e^4,f^d=f^10,g^d=g^10

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36300, 481);