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ΓL(1,𝔽8)×#11717375 #72243174

order16800
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productΓL(1,𝔽8)×#11717375

Splittings

FittingC22×C52×C7 ⋊ C2×C3×C4
HallC7×#11717375 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C4×D4
HallC52×C7 ⋊ C3×C4×D4
SylowC52 ⋊ C4×D4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×#11717375

Subgroups

subgroupG/H
center, FrattiniC2C2×ΓL(1,𝔽8)×D5⋊AGL(1,𝔽5)
commutator, socleC2×C52×C7C22×C3×C4
FittingC22×C52×C7C2×C3×C4
prime radicalΓL(1,𝔽8)×#87886C2
cosocleC2×C7×C5⋊D5C23×C3
hypercenterC22ΓL(1,𝔽8)×D5⋊AGL(1,𝔽5)
hypocenterC52×C7C3×C4×D4
LCSG ▷ C2×C52×C7 ▷ C52×C7
UCSC1 ◁ C2 ◁ C22

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#11717375
maximal normal subgroup ofAGL(1,𝔽7)×#11717375

GAP

Past the SmallGroups library coverage — no SmallGroups id.