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C2×C3×C5⋊(ΓL(1,𝔽8)⋊D23) #44394525

order28980
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C5⋊(ΓL(1,𝔽8)⋊D23)
unambiguous semidirect product(C2×C5)⋊(C3×ΓL(1,𝔽8)⋊D23)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C3×D115)
unambiguous semidirect product(C2×C23)⋊(C3×C5⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×D805)⋊C32
unambiguous semidirect product(C2×C3×D805)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C23 ⋊ C2×C3
derived, HallC5×C7×C23 ⋊ C22×C32
HallC3×C5×ΓL(1,𝔽8)×C23 ⋊ C22
HallC2×D805 ⋊ C32
HallC3×ΓL(1,𝔽8)×C23 ⋊ C2×D5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D23
HallC7×C23 ⋊ C2×C32×D5
HallC5×C23 ⋊ C2×C3×AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×D115
HallC5×C7 ⋊ C2×C32×D23
SylowC23 ⋊ C2×C3×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C32×D115
SylowC5 ⋊ C2×C3×ΓL(1,𝔽8)⋊D23

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊(ΓL(1,𝔽8)⋊D23)
commutator, cosocle, hypocenterC5×C7×C23C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C23C2×C3
LCSG ▷ C5×C7×C23
UCSC1 ◁ C2×C3

240 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 4830

Conjugacy classes & character table

1752 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28980, 194);