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Q8×AGL(1,𝔽5)×AGL(1,𝔽7) #72218434

order6720
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productQ8×AGL(1,𝔽5)×AGL(1,𝔽7)

Splittings

FittingC5×C7×Q8 ⋊ C2×C3×C4
HallQ8×D7×AGL(1,𝔽5) ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×C4×Q8
HallC5×C7 ⋊ C2×C3×C4×Q8
HallΓL(1,𝔽8) ⋊ C2×Q8×AGL(1,𝔽5)
SylowC7 ⋊ C2×C3×Q8×AGL(1,𝔽5)
SylowC5 ⋊ C4×Q8×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, FrattiniC2C22×AGL(1,𝔽5)×AGL(1,𝔽7)
commutator, socleC2×C5×C7C23×C3×C4
FittingC5×C7×Q8C2×C3×C4
prime radicalC2×D5×AGL(1,𝔽7)C23
cosocleC2×C7×D5C24×C3
hypercenterQ8AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×C4×Q8
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ Q8

1192 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#65735, SD16×AGL(1,𝔽5)×AGL(1,𝔽7), #41×AGL(1,𝔽5)×AGL(1,𝔽7), …
maximal normal subgroup ofAGL(1,𝔽7)×#65735, SD16×AGL(1,𝔽5)×AGL(1,𝔽7), #41×AGL(1,𝔽5)×AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.