Finite Group Databasebeta

← Prev (#17987697) Next (#17987699) →

C7×C11×ΓL(1,𝔽16) #17987698

order4620
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C4×C7×C11 · Schur multiplier (H2) = C1

Aliases

direct productC7×C11×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C7×C11×AGL(1,𝔽5))

Splittings

Fitting, socle, HallC3×C5×C7×C11 ⋊ C4
derived, HallC3×C5 ⋊ C4×C7×C11
HallC5×C7×C11 ⋊ Dic3
HallC3×C7×C11 ⋊ AGL(1,𝔽5)
HallC3×C5×C11 ⋊ C4×C7
HallC3×C5×C7 ⋊ C4×C11
HallC5×C11 ⋊ C7×Dic3
HallC5×C7 ⋊ C11×Dic3
HallC3×C11 ⋊ C7×AGL(1,𝔽5)
HallC3×C7 ⋊ C11×AGL(1,𝔽5)
SylowC5 ⋊ C7×C11×Dic3
SylowC3 ⋊ C7×C11×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC7×C11ΓL(1,𝔽16)
commutator, hypocenterC3×C5C4×C7×C11
FrattiniC1G
Fitting, socleC3×C5×C7×C11C4
prime radicalC3×C7×D5×C11C2
cosocleC3×D5C2×C7×C11
LCSG ▷ C3×C5
UCSC1 ◁ C7×C11

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 1155

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(20,21,22,23)(25,26);(19,20,23,21,22)(24,25,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×C11×C9⋊AGL(1,𝔽5), C11×ΓL(1,𝔽8)×ΓL(1,𝔽16), C3×C7×C11×ΓL(1,𝔽16), …
maximal normal subgroup ofC11×ΓL(1,𝔽8)×ΓL(1,𝔽16), C3×C7×C11×ΓL(1,𝔽16), C7×C11⋊C5×ΓL(1,𝔽16), …
maximal quotient ofC7×C11×C9⋊AGL(1,𝔽5), C3×C7×C11×ΓL(1,𝔽16), C7×C11×C3⋊ΓL(1,𝔽16), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4620, 95);