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C2×AGL(1,𝔽7)×C22⋊AGL(1,𝔽5) #72217880

order6720
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×AGL(1,𝔽7)×C22⋊AGL(1,𝔽5)

Splittings

FittingC23×C5×C7 ⋊ C2×C3×C4
HallC2×D7×C22⋊AGL(1,𝔽5) ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C22×C22⋊C4
HallC5×C7 ⋊ C22×C3×C22⋊C4
HallΓL(1,𝔽8) ⋊ C22×C22⋊AGL(1,𝔽5)
SylowC7 ⋊ C22×C3×C22⋊AGL(1,𝔽5)
SylowC5 ⋊ C2×C22⋊C4×AGL(1,𝔽7)

Subgroups

subgroupG/H
centerC22C2×AGL(1,𝔽5)×AGL(1,𝔽7)
commutatorC2×C5×C7C23×C3×C4
FrattiniC2C22×AGL(1,𝔽5)×AGL(1,𝔽7)
FittingC23×C5×C7C2×C3×C4
socleC22×C5×C7C22×C3×C4
prime radicalC23×D5×AGL(1,𝔽7)C2
cosocleC2×C7×D5C24×C3
hypercenterC23AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C22×C3×C22⋊C4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C23

2696 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×AGL(1,𝔽5)×AGL(1,𝔽7), C22⋊C4×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽7)×#65699, …
maximal normal subgroup ofC2×D4×AGL(1,𝔽5)×AGL(1,𝔽7), C22⋊C4×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽7)×#65699, …
maximal quotient ofAGL(1,𝔽7)×#65767, AGL(1,𝔽7)×#65793, AGL(1,𝔽7)×#65800, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.