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C7×C29×ΓL(1,𝔽16) #27131780

order12180
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C7×C29 · Schur multiplier (H2) = C1

Aliases

direct productC7×C29×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C7×AGL(1,𝔽5)×C29)

Splittings

Fitting, socle, HallC3×C5×C7×C29 ⋊ C4
derived, HallC3×C5 ⋊ C4×C7×C29
HallC5×C7×C29 ⋊ Dic3
HallC3×C7×C29 ⋊ AGL(1,𝔽5)
HallC3×C5×C29 ⋊ C4×C7
HallC5×C29 ⋊ C7×Dic3
HallC3×C5×C7 ⋊ C4×C29
HallC3×C29 ⋊ C7×AGL(1,𝔽5)
HallC5×C7 ⋊ Dic3×C29
HallC3×C7 ⋊ AGL(1,𝔽5)×C29
SylowC5 ⋊ C7×Dic3×C29
SylowC3 ⋊ C7×AGL(1,𝔽5)×C29

Subgroups

subgroupG/H
center, hypercenterC7×C29ΓL(1,𝔽16)
commutator, hypocenterC3×C5C4×C7×C29
FrattiniC1G
Fitting, socleC3×C5×C7×C29C4
LCSG ▷ C3×C5
UCSC1 ◁ C7×C29

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1827 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×C29×C9⋊AGL(1,𝔽5), ΓL(1,𝔽8)×C29×ΓL(1,𝔽16), C3×C7×C29×ΓL(1,𝔽16), …
maximal normal subgroup ofΓL(1,𝔽8)×C29×ΓL(1,𝔽16), C3×C7×C29×ΓL(1,𝔽16)
maximal quotient ofC7×C29×C9⋊AGL(1,𝔽5), C3×C7×C29×ΓL(1,𝔽16), C7×C29×C3⋊ΓL(1,𝔽16)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12180, 115);