Finite Group Databasebeta

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C22×C5×C7×C11×ΓL(1,𝔽8) #44568929

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5×C7×C11 · Schur multiplier (H2) = C2

Aliases

direct productC22×C5×C7×C11×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C5×C72×C11 ⋊ C3
derivedC7 ⋊ C22×C3×C5×C7×C11
HallC5×C72×C11 ⋊ C22×C3
HallC22×C72×C11 ⋊ C3×C5
HallC22×C5×C72 ⋊ C3×C11
HallC72×C11 ⋊ C22×C3×C5
HallC5×C72 ⋊ C22×C3×C11
HallC22×C72 ⋊ C3×C5×C11
SylowC72 ⋊ C22×C3×C5×C11

Subgroups

subgroupG/H
center, hypercenterC22×C5×C7×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C5×C7×C11
FrattiniC1G
Fitting, socleC22×C5×C72×C11C3
LCSG ▷ C7
UCSC1 ◁ C22×C5×C7×C11

200 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 4620

Conjugacy classes & character table

7700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 631);