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C22×C4×AGL(1,𝔽7) #10621508

order672
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×C4×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C22×C3×C4)

Splittings

FittingC22×C4×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C23×C3×C4
HallC22×C4×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C23×C4

Subgroups

subgroupG/H
center, hypercenterC22×C4AGL(1,𝔽7)
commutator, hypocenterC7C23×C3×C4
FrattiniC2C23×AGL(1,𝔽7)
FittingC22×C4×C7C2×C3
socleC23×C7C22×C3
prime radicalC23×AGL(1,𝔽7)C2
cosocleC2×C7C24×C3
LCSG ▷ C7
UCSC1 ◁ C22×C4

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89438⋊C3, #89439⋊C3, #95×AGL(1,𝔽7), …
maximal normal subgroup of#89438⋊C3, #89439⋊C3, #95×AGL(1,𝔽7), …
maximal quotient ofC2×C4×ΓL(1,𝔽8)⋊Q8, C2×C42×AGL(1,𝔽7), C2×#6435⋊C3, …
automorphism group ofD7×C16, #6372, C5×C7⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 1090);