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(C3⋊AGL(1,𝔽7))⋊D19 #18012245

order4788
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C3⋊AGL(1,𝔽7))⋊D19
unambiguous semidirect product(ΓL(1,𝔽8)⋊D19)⋊D3

Splittings

Fitting, socle, derivedC3×C7×C19 ⋊ C22×C3
HallC3×C19×ΓL(1,𝔽8) ⋊ C22
HallC7×C19 ⋊ C2×C3×D3
HallC3×ΓL(1,𝔽8) ⋊ C2×D19
SylowC19 ⋊ D3×AGL(1,𝔽7)
SylowC7 ⋊ C3×D3×D19

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C19C22×C3
LCSG ▷ C3×C7×C19
UCSC1

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 399

Polycyclic presentation:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×(C3⋊AGL(1,𝔽7))⋊D19, (ΓL(1,𝔽8)⋊D19)⋊D15, #37561385, …
maximal normal subgroup ofC5×(C3⋊AGL(1,𝔽7))⋊D19, C7×(C3⋊AGL(1,𝔽7))⋊D19
maximal quotient ofC5×(C3⋊AGL(1,𝔽7))⋊D19, (ΓL(1,𝔽8)⋊D19)⋊D15, #37561385, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 161);