Finite Group Databasebeta

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C2×ΓL(1,𝔽8)×D455 #50304299

order38220
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)×D455
unambiguous semidirect product(C2×C5)⋊(ΓL(1,𝔽8)×D91)

Splittings

Fitting, socleC2×C5×C72×C13 ⋊ C2×C3
derived, HallC5×C72×C13 ⋊ C22×C3
HallC2×C7×D455 ⋊ C3
HallC5×C7×C13×ΓL(1,𝔽8) ⋊ C22
HallC7×C13×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×D13
HallC72×C13 ⋊ C2×C3×D5
HallC5×C72 ⋊ C2×C3×D13
HallC7×ΓL(1,𝔽8) ⋊ C2×D65
HallC5×C13 ⋊ C2×D7×ΓL(1,𝔽8)
SylowC72 ⋊ C2×C3×D65
SylowC13 ⋊ C2×ΓL(1,𝔽8)×D35
SylowC5 ⋊ C2×ΓL(1,𝔽8)×D91

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D455
commutator, cosocle, hypocenterC5×C72×C13C22×C3
FrattiniC1G
Fitting, socleC2×C5×C72×C13C2×C3
LCSG ▷ C5×C72×C13
UCSC1 ◁ C2

168 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 5460

Conjugacy classes & character table

2290 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 985);