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C3⋊((C19⋊C3)⋊AGL(1,𝔽5)) #16811883

order3420
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊((C19⋊C3)⋊AGL(1,𝔽5))
unambiguous semidirect product(C19⋊C3)⋊ΓL(1,𝔽16)
unambiguous semidirect product(C3⋊(C19⋊AGL(1,𝔽5)))⋊C3

Splittings

Fitting, socle, derivedC3×C5×C19 ⋊ C3×C4
HallC3×C5×C19⋊C3 ⋊ C4
HallC3×C19⋊C3 ⋊ AGL(1,𝔽5)
HallC5×C19 ⋊ C3×Dic3
SylowC19 ⋊ C3×ΓL(1,𝔽16)
SylowC5 ⋊ C3⋊((C19⋊C3)⋊C4)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C19C3×C4
prime radicalC3×D5×C19⋊C3C2
cosocleC3×D5×C19C2×C3
LCSG ▷ C3×C5×C19
UCSC1

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 285

Polycyclic presentation:

2,3,2,3,5,19|a^2=c,d^a=d^2,e^a=e^2,f^a=f^18,f^b=f^7,e^c=e^4

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊((C19⋊C3)⋊#1035), C5×C3⋊((C19⋊C3)⋊AGL(1,𝔽5)), C3⋊((C19⋊C3)⋊#1043), …
maximal normal subgroup ofC5×C3⋊((C19⋊C3)⋊AGL(1,𝔽5)), C7×C3⋊((C19⋊C3)⋊AGL(1,𝔽5)), C11×C3⋊((C19⋊C3)⋊AGL(1,𝔽5))
maximal quotient ofC3⋊((C19⋊C3)⋊#1035), C5×C3⋊((C19⋊C3)⋊AGL(1,𝔽5)), C3⋊((C19⋊C3)⋊#1043), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3420, 76);