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C2×D4×AGL(1,𝔽5)×ΓL(1,𝔽8) #72217841

order6720
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×D4×AGL(1,𝔽5)×ΓL(1,𝔽8)

Splittings

FittingC2×C5×C7×D4 ⋊ C3×C4
HallC2×C7×D4×AGL(1,𝔽5) ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×C4×D4
HallC5×C7 ⋊ C2×C3×C4×D4
SylowC7 ⋊ C2×C3×D4×AGL(1,𝔽5)
SylowC5 ⋊ C2×C4×D4×ΓL(1,𝔽8)

Subgroups

subgroupG/H
centerC22C22×AGL(1,𝔽5)×ΓL(1,𝔽8)
commutatorC2×C5×C7C23×C3×C4
FrattiniC2C23×AGL(1,𝔽5)×ΓL(1,𝔽8)
FittingC2×C5×C7×D4C3×C4
socleC22×C5×C7C22×C3×C4
prime radicalC2×D4×D5×ΓL(1,𝔽8)C2
cosocleC2×C7×D5C24×C3
hypercenterC2×D4AGL(1,𝔽5)×ΓL(1,𝔽8)
hypocenterC5×C7C2×C3×C4×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C2×D4

1704 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,ℤ/8), AGL(1,𝔽5)×ΓL(1,𝔽8)×#120, …
maximal normal subgroup ofC2×D4×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,ℤ/8), AGL(1,𝔽5)×ΓL(1,𝔽8)×#120, …
maximal quotient ofAGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,ℤ/8), AGL(1,𝔽5)×ΓL(1,𝔽8)×#120

GAP

Past the SmallGroups library coverage — no SmallGroups id.