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(C5⋊AGL(1,𝔽7))⋊(D19⋊C3) #37561212

order23940
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

unambiguous semidirect product(C5⋊AGL(1,𝔽7))⋊(D19⋊C3)
unambiguous semidirect product(C5⋊(D19⋊C3))⋊AGL(1,𝔽7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊(D19⋊C3))⋊D5
unambiguous semidirect product(#11716327⋊D5)⋊C3
unambiguous semidirect product(#11716328⋊D5)⋊C3
unambiguous semidirect product((C5⋊AGL(1,𝔽7))⋊D19)⋊C3
unambiguous semidirect product((C5⋊(D19⋊C3))⋊D7)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C19 ⋊ C22×C32
HallC5×ΓL(1,𝔽8)×C19⋊C3 ⋊ C22
HallD35⋊D19 ⋊ C32
HallΓL(1,𝔽8)×C19⋊C3 ⋊ C2×D5
HallC7×C19 ⋊ C2×C32×D5
HallC5×C19 ⋊ C2×C3×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×C3×D19⋊C3
SylowC19 ⋊ C3×D5×AGL(1,𝔽7)
SylowC7 ⋊ C3×D5×D19⋊C3
SylowC5 ⋊ AGL(1,𝔽7)×D19⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C19C22×C32
LCSG ▷ C5×C7×C19
UCSC1

240 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 665

Polycyclic presentation:

2^2,3^2,5,7,19|f^a=f^6,g^a=g^18,e^b=e^4,f^b=f^6,g^c=g^7,f^d=f^2

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 344);