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C2×D5×AGL(1,𝔽7) #11719841

order840
isoclinismisoclinic class of D5×AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×D5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×D5)
unambiguous semidirect product(D5×D7)⋊(C2×C3)
unambiguous semidirect product(C5×AGL(1,𝔽7))⋊C22
unambiguous semidirect product(D5×ΓL(1,𝔽8))⋊C22
unambiguous semidirect product(C5⋊AGL(1,𝔽7))⋊C22
unambiguous semidirect product(C2×D5×D7)⋊C3

Splittings

Fitting, socleC2×C5×C7 ⋊ C22×C3
derived, HallC5×C7 ⋊ C23×C3
HallC2×D5×D7 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C23
HallΓL(1,𝔽8) ⋊ C22×D5
SylowC7 ⋊ C22×C3×D5
SylowC5 ⋊ C22×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2D5×AGL(1,𝔽7)
commutator, cosocle, hypocenterC5×C7C23×C3
FrattiniC1G
Fitting, socleC2×C5×C7C22×C3
LCSG ▷ C5×C7
UCSC1 ◁ C2

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 70

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊AGL(1,𝔽7), C4×D5×AGL(1,𝔽7), D20×AGL(1,𝔽7), …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊AGL(1,𝔽7), C4×D5×AGL(1,𝔽7), D20×AGL(1,𝔽7), …
maximal quotient ofDic10×AGL(1,𝔽7), #10587938⋊C3, #10587939⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 143);