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C2×D7×ΓL(1,𝔽8)×C11⋊C5 #44568573

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

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2D7×ΓL(1,𝔽8)×C11⋊C5
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 63 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 275);