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C3⋊(C3⋊ΓL(1,𝔽16)) #10587432

order540
isoclinismisoclinic class of C3⋊(C3⋊ΓL(1,𝔽16))
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C33

Aliases

unambiguous semidirect productC3⋊(C3⋊ΓL(1,𝔽16))

Splittings

Fitting, socle, derived, HallC33×C5 ⋊ C4
SylowC33 ⋊ AGL(1,𝔽5)
SylowC5 ⋊ C3⋊(C3⋊Dic3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC33×C5C4
prime radical, cosocleC33×D5C2
LCSG ▷ C33×C5
UCSC1

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 135

Polycyclic presentation:

2^2,3^3,5|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^3,f^b=f^4

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C3⋊(C3⋊D3), D3×C3⋊ΓL(1,𝔽16), C3⋊D3×ΓL(1,𝔽16), …
maximal normal subgroup ofAGL(1,𝔽5)×C3⋊(C3⋊D3), D3×C3⋊ΓL(1,𝔽16), C3⋊D3×ΓL(1,𝔽16), …
maximal quotient ofC3⋊(C3⋊(C3⋊#170)), C2×C3⋊(C3⋊ΓL(1,𝔽16)), C3⋊(C3⋊(C9⋊AGL(1,𝔽5))), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 92);