Finite Group Databasebeta

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C5×AGL(1,𝔽7)×D55 #37528518

order23100
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×AGL(1,𝔽7)×D55
unambiguous semidirect productD7⋊(C3×C5×D55)
unambiguous semidirect product(C5×AGL(1,𝔽7))⋊(C5×D11)
unambiguous semidirect product(C5×D7×D55)⋊C3

Splittings

Fitting, socle, HallC52×C7×C11 ⋊ C22×C3
derivedC5×C7×C11 ⋊ C22×C3×C5
HallC5×D7×D55 ⋊ C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C22
HallC52×ΓL(1,𝔽8) ⋊ C2×D11
HallC52×C11 ⋊ C2×AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C2×C5×D5
HallC52×C7 ⋊ C2×C3×D11
HallC7×C11 ⋊ C2×C3×C5×D5
SylowC52 ⋊ D11×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×C5×D55
SylowC11 ⋊ C5×D5×AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×C5×D55

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽7)×D55
commutator, cosocle, hypocenterC5×C7×C11C22×C3×C5
FrattiniC1G
Fitting, socleC52×C7×C11C22×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C5

192 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 770

Conjugacy classes & character table

1015 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 329);