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C2×C8×AGL(1,𝔽7) #10620735

order672
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×C8×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×C8)
unambiguous semidirect product(C2×C8×D7)⋊C3

Splittings

FittingC2×C7×C8 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C3×C8
HallC2×C8×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC2×C8AGL(1,𝔽7)
commutator, hypocenterC7C22×C3×C8
FrattiniC4C22×AGL(1,𝔽7)
FittingC2×C7×C8C2×C3
socleC22×C7C2×C3×C4
prime radicalC22×AGL(1,𝔽7)C4
cosocleC4×C7C23×C3
LCSG ▷ C7
UCSC1 ◁ C2×C8

152 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(C2×AGL(1,𝔽7))⋊C16, C4×C8×AGL(1,𝔽7), C8×C4⋊AGL(1,𝔽7), …
maximal normal subgroup of(C2×AGL(1,𝔽7))⋊C16, C4×C8×AGL(1,𝔽7), C8×C4⋊AGL(1,𝔽7), …
maximal quotient ofC8×ΓL(1,𝔽8)⋊Q8, C4×C8×AGL(1,𝔽7), #89457⋊C3, …
automorphism group ofC7⋊C32

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 317);