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C53×C3⋊AGL(1,𝔽7) #20269247

order6678
isoclinismisoclinic class of C3⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C53 · Schur multiplier (H2) = C1

Aliases

direct productC53×C3⋊AGL(1,𝔽7)
unambiguous semidirect productC3⋊(AGL(1,𝔽7)×C53)
unambiguous semidirect productΓL(1,𝔽8)⋊(D3×C53)
unambiguous semidirect productD21⋊(C3×C53)
unambiguous semidirect product(D21×C53)⋊C3

Splittings

Fitting, socleC3×C7×C53 ⋊ C2×C3
derivedC3×C7 ⋊ C2×C3×C53
HallC3×ΓL(1,𝔽8)×C53 ⋊ C2
HallC7×C53 ⋊ C3×D3
HallC3×ΓL(1,𝔽8) ⋊ C2×C53
SylowC7 ⋊ C3×D3×C53

Subgroups

subgroupG/H
center, hypercenterC53C3⋊AGL(1,𝔽7)
commutator, hypocenterC3×C7C2×C3×C53
FrattiniC1G
Fitting, socleC3×C7×C53C2×C3
LCSG ▷ C3×C7
UCSC1 ◁ C53

36 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

636 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD53×C3⋊AGL(1,𝔽7), D3×AGL(1,𝔽7)×C53, (C3⋊AGL(1,𝔽7))⋊D53, …
maximal normal subgroup ofD53×C3⋊AGL(1,𝔽7), D3×AGL(1,𝔽7)×C53, (C3⋊AGL(1,𝔽7))⋊D53, …
maximal quotient ofC53×C3⋊(ΓL(1,𝔽8)⋊C4), C2×C53×C3⋊AGL(1,𝔽7), C5×C53×C3⋊AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6678, 17);