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C22×C7×ΓL(1,𝔽16) #13132964

order1680
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23

Aliases

direct productC22×C7×ΓL(1,𝔽16)
unambiguous semidirect productD5⋊(C2×C7×Dic3)

Splittings

Fitting, socleC22×C3×C5×C7 ⋊ C4
derived, HallC3×C5 ⋊ C22×C4×C7
HallC3×C5×C7 ⋊ C22×C4
HallC5×C7 ⋊ C22×Dic3
HallC3×C7 ⋊ C22×AGL(1,𝔽5)
SylowC5 ⋊ C22×C7×Dic3
SylowC3 ⋊ C22×C7×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC22×C7ΓL(1,𝔽16)
commutator, hypocenterC3×C5C22×C4×C7
FrattiniC1G
Fitting, socleC22×C3×C5×C7C4
prime radicalC22×C3×C7×D5C2
cosocleC3×D5C23×C7
LCSG ▷ C3×C5
UCSC1 ◁ C22×C7

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×D3×C7×AGL(1,𝔽5), C2×C4×C7×ΓL(1,𝔽16), C2×C7×Dic3×AGL(1,𝔽5)
maximal normal subgroup ofC22×D3×C7×AGL(1,𝔽5), C2×C4×C7×ΓL(1,𝔽16), C2×C7×Dic3×AGL(1,𝔽5)
maximal quotient ofC2×C4×C7×ΓL(1,𝔽16)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 984);