Finite Group Databasebeta

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C2×ΓL(1,𝔽8)×C7⋊AGL(1,𝔽11) #44568572

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

Aliases

direct productC2×ΓL(1,𝔽8)×C7⋊AGL(1,𝔽11)
unambiguous semidirect product(C2×C11⋊C5)⋊(D7×ΓL(1,𝔽8))
unambiguous semidirect product(C2×ΓL(1,𝔽8)×D77)⋊C5

Splittings

Fitting, socleC2×C72×C11 ⋊ C2×C3×C5
derived, HallC72×C11 ⋊ C22×C3×C5
HallC2×C7×C7⋊AGL(1,𝔽11) ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC2×ΓL(1,𝔽8)×D77 ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
HallC2×C7×D77 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC7×ΓL(1,𝔽8) ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×D7×ΓL(1,𝔽8)
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×D7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×C7⋊AGL(1,𝔽11)
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1 ◁ C2

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

normal subgroups only

This group has 168 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) = 924

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 274);