Finite Group Databasebeta

← Prev (#11929533) Next (#11929535) →

C7×C3⋊ΓL(1,𝔽16) #11929534

order1260
isoclinismisoclinic class of C3⋊ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C7×ΓL(1,𝔽16))

Splittings

Fitting, socle, HallC32×C5×C7 ⋊ C4
derived, HallC32×C5 ⋊ C4×C7
HallC32×C7 ⋊ AGL(1,𝔽5)
HallC5×C7 ⋊ C3⋊Dic3
SylowC32 ⋊ C7×AGL(1,𝔽5)
SylowC5 ⋊ C7×C3⋊Dic3

Subgroups

subgroupG/H
center, hypercenterC7C3⋊ΓL(1,𝔽16)
commutator, hypocenterC32×C5C4×C7
FrattiniC1G
Fitting, socleC32×C5×C7C4
prime radicalC32×C7×D5C2
cosocleC32×D5C2×C7
LCSG ▷ C32×C5
UCSC1 ◁ C7

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 315

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2,3,4,5,6,7)(9,10,11,12)(14,15)(17,18);(8,9,12,10,11)(16,17,18);(13,14,15)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×C11×C3⋊ΓL(1,𝔽16), C7×C3⋊(C3⋊(C11⋊AGL(1,𝔽5))), C7×C13×C3⋊ΓL(1,𝔽16), …
maximal normal subgroup ofC7×C11×C3⋊ΓL(1,𝔽16), C7×C13×C3⋊ΓL(1,𝔽16), C7×C17×C3⋊ΓL(1,𝔽16), …
maximal quotient ofC7×C11×C3⋊ΓL(1,𝔽16), C7×C3⋊(C3⋊(C11⋊AGL(1,𝔽5))), C7×C13×C3⋊ΓL(1,𝔽16), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 86);