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C2×C5×C3⋊(ΓL(1,𝔽8)⋊D19) #37561472

order23940
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×C3⋊(ΓL(1,𝔽8)⋊D19)
unambiguous semidirect product(C2×C3)⋊(C5×ΓL(1,𝔽8)⋊D19)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C5×D57)

Splittings

Fitting, socleC2×C3×C5×C7×C19 ⋊ C2×C3
derivedC3×C7×C19 ⋊ C22×C3×C5
HallC3×C5×C19×ΓL(1,𝔽8) ⋊ C22
HallC3×C19×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7×C19 ⋊ C2×C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D19
HallC7×C19 ⋊ C2×C3×C5×D3
HallC5×C19 ⋊ C2×C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×C5×D19
HallC5×C7 ⋊ C2×C3×D57
SylowC19 ⋊ C2×C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×C5×D57

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊(ΓL(1,𝔽8)⋊D19)
commutator, cosocle, hypocenterC3×C7×C19C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C19C2×C3
LCSG ▷ C3×C7×C19
UCSC1 ◁ C2×C5

176 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 3990

Conjugacy classes & character table

1470 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 604);