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C2×C3×AGL(1,𝔽5)×AGL(1,𝔽7) #72217343

order5040
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C3×AGL(1,𝔽5)×AGL(1,𝔽7)

Splittings

Fitting, socleC2×C3×C5×C7 ⋊ C2×C3×C4
derived, HallC5×C7 ⋊ C22×C32×C4
HallC2×D7×AGL(1,𝔽5) ⋊ C32
HallC3×C5×ΓL(1,𝔽8) ⋊ C22×C4
HallC3×ΓL(1,𝔽8) ⋊ C22×AGL(1,𝔽5)
SylowC7 ⋊ C22×C32×AGL(1,𝔽5)
SylowC5 ⋊ C2×C3×C4×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,𝔽5)×AGL(1,𝔽7)
commutator, hypocenterC5×C7C22×C32×C4
FrattiniC1G
Fitting, socleC2×C3×C5×C7C2×C3×C4
prime radicalC2×C3×D5×AGL(1,𝔽7)C2
cosocleC7×D5C23×C32
LCSG ▷ C5×C7
UCSC1 ◁ C2×C3

648 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×AGL(1,𝔽5)×AGL(1,𝔽7), C3×D3×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽5)×SL(2,𝔽3)×AGL(1,𝔽7), …
maximal normal subgroup ofC2×D3×AGL(1,𝔽5)×AGL(1,𝔽7)
maximal quotient ofC3×D3×AGL(1,𝔽5)×AGL(1,𝔽7), C2×AGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), AGL(1,𝔽5)×AGL(1,𝔽7)2, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.