Finite Group Databasebeta

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C7×ΓL(1,𝔽8)×Dic65 #50303620

order38220
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)×Dic65

Splittings

derivedC5×C7×C13 ⋊ C3×C4×C7
HallC72×Dic65 ⋊ C3
HallC5×C7×C13×ΓL(1,𝔽8) ⋊ C4
HallC5×C72×C13 ⋊ C3×C4
HallC7×C13×ΓL(1,𝔽8) ⋊ Dic5
HallC5×C7×ΓL(1,𝔽8) ⋊ Dic13
HallC72×C13 ⋊ C3×Dic5
HallC5×C72 ⋊ C3×Dic13
HallC5×C13 ⋊ C4×C7×ΓL(1,𝔽8)
SylowC72 ⋊ C3×Dic65
SylowC13 ⋊ C7×Dic5×ΓL(1,𝔽8)
SylowC5 ⋊ C7×ΓL(1,𝔽8)×Dic13

Subgroups

subgroupG/H
center, hypercenterC2×C7ΓL(1,𝔽8)×D65
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC2C7×ΓL(1,𝔽8)×D65
Fitting, socleC2×C5×C72×C13C2×C3
prime radicalC2×C5×C7×C13×ΓL(1,𝔽8)C2
cosocleC2×C5×C7×C13C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2×C7

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

normal subgroups only

This group has 120 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) = 5460

Conjugacy classes & character table

2380 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 306);