Finite Group Databasebeta

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C5×D5×ΓL(1,𝔽8)×D11 #37528521

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

Aliases

direct productC5×D5×ΓL(1,𝔽8)×D11
unambiguous semidirect product(C5×C7×D5×D11)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC5D5×ΓL(1,𝔽8)×D11
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 60 normal subgroups are shown.

normal subgroups only

This group has 192 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) = 770

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 332);