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C22×C5×AGL(1,𝔽7) #11719854

order840
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C23

Aliases

direct productC22×C5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C22×C3×C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C23×C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C22×C5)

Splittings

Fitting, socleC22×C5×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C23×C3×C5
HallC22×C5×D7 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C23
HallC22×D7 ⋊ C3×C5
HallC5×C7 ⋊ C23×C3
HallΓL(1,𝔽8) ⋊ C23×C5

Subgroups

subgroupG/H
center, hypercenterC22×C5AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C23×C3×C5
FrattiniC1G
Fitting, socleC22×C5×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C22×C5

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽7))⋊Dic5, C5×(C2×AGL(1,𝔽7))⋊C4, C2×Dic5×AGL(1,𝔽7), …
maximal normal subgroup of(C2×AGL(1,𝔽7))⋊Dic5, C5×(C2×AGL(1,𝔽7))⋊C4, C2×Dic5×AGL(1,𝔽7), …
maximal quotient ofC2×C5×ΓL(1,𝔽8)⋊Q8, C2×C4×C5×AGL(1,𝔽7), C2×C5×C4⋊AGL(1,𝔽7), …
automorphism group ofC11×Dic7, C2×C11×D7, C3×C11×D7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 156);