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ΓL(1,𝔽8)×#1313 #72276538

order2688
automorphism?
homologyAbelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#1313

Splittings

Fitting, HallC7×#1313 ⋊ C3
SylowC7 ⋊ C3×#1313

Subgroups

subgroupG/H
centerC2×C4C42×ΓL(1,𝔽8)
commutatorC4×C7C3×C4×C8
FrattiniC4×C8C22×ΓL(1,𝔽8)
FittingC7×#1313C3
socleC22×C7C3×#97
prime radicalC22×ΓL(1,𝔽8)#97
cosocleC4×C7×C8C22×C3
hypercenter#1313ΓL(1,𝔽8)
hypocenterC7C3×#1313
LCSG ▷ C4×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #1313

160 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 896

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#1313
maximal normal subgroup ofAGL(1,𝔽7)×#1313

GAP

Past the SmallGroups library coverage — no SmallGroups id.