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(C3⋊ΓL(1,𝔽16))⋊C3 #10587377

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

Aliases

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

Splittings

Fitting, HallC5×H(3,𝔽3) ⋊ C4
derivedC32×C5 ⋊ C3×C4
SylowH(3,𝔽3) ⋊ AGL(1,𝔽5)
SylowC5 ⋊ (C3⋊Dic3)⋊C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC32×C5C3×C4
FrattiniC3C3×ΓL(1,𝔽16)
FittingC5×H(3,𝔽3)C4
socleC3×C5C3×Dic3
prime radicalD5×H(3,𝔽3)C2
cosocleC32×D5C2×C3
LCSG ▷ C32×C5
UCSC1

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 45

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×(C3⋊D3)⋊C3, ((C3⋊D3)⋊C3)⋊AGL(1,𝔽5), #11761932, …
maximal normal subgroup ofAGL(1,𝔽5)×(C3⋊D3)⋊C3, ((C3⋊D3)⋊C3)⋊AGL(1,𝔽5), #11761932, …
maximal quotient of(C3⋊(C3⋊#170))⋊C3, C2×(C3⋊ΓL(1,𝔽16))⋊C3, (C3⋊ΓL(1,𝔽16))⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 37);