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AGL(1,𝔽7)×#442 #72218546

order2688
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productAGL(1,𝔽7)×#442

Splittings

FittingC7×#442 ⋊ C2×C3
HallD7×#442 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#442
SylowC7 ⋊ C2×C3×#442

Subgroups

subgroupG/H
centerC22C2×D4×AGL(1,𝔽7)
commutatorC4×C7C23×C3×C4
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC7×#442C2×C3
socleC22×C7C22×C3×D4
prime radicalC2×D8×AGL(1,𝔽7)C2
cosocleC2×C4×C7C24×C3
hypercenter#442AGL(1,𝔽7)
hypocenterC7C2×C3×#442
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #442

1216 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#65731, C4×AGL(1,ℤ/8)×AGL(1,𝔽7), AGL(1,𝔽7)×AGL(1,ℤ/16), …
maximal normal subgroup ofC4×AGL(1,ℤ/8)×AGL(1,𝔽7), AGL(1,𝔽7)×AGL(1,ℤ/16), AGL(1,𝔽7)×#3184
maximal quotient ofAGL(1,𝔽7)×#65731, AGL(1,𝔽7)×AGL(1,ℤ/16)
automorphism group of#6374, #6400, #6401, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.