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C11×C5⋊AGL(1,𝔽7) #14556348

order2310
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×C5⋊AGL(1,𝔽7)
unambiguous semidirect productC5⋊(C11×AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(D5×C11)
unambiguous semidirect productD35⋊(C3×C11)
unambiguous semidirect product(C11×D35)⋊C3

Splittings

Fitting, socle, HallC5×C7×C11 ⋊ C2×C3
derived, HallC5×C7 ⋊ C2×C3×C11
HallC5×C11×ΓL(1,𝔽8) ⋊ C2
HallC11×D35 ⋊ C3
HallC11×ΓL(1,𝔽8) ⋊ D5
HallC5×ΓL(1,𝔽8) ⋊ C2×C11
HallC7×C11 ⋊ C3×D5
HallD35 ⋊ C3×C11
HallC5×C11 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D5×C11
SylowC7 ⋊ C3×D5×C11
SylowC5 ⋊ C11×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC11C5⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC5×C7C2×C3×C11
FrattiniC1G
Fitting, socleC5×C7×C11C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C11

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 385

A minimal faithful permutation representation — 2 generators on 23 points:

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

Conjugacy classes & character table

187 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×C5⋊AGL(1,𝔽7), D5×C11×AGL(1,𝔽7), D55⋊AGL(1,𝔽7), …
maximal normal subgroup ofD11×C5⋊AGL(1,𝔽7), D5×C11×AGL(1,𝔽7), D55⋊AGL(1,𝔽7), …
maximal quotient ofC11×ΓL(1,𝔽8)⋊Dic5, C2×C11×C5⋊AGL(1,𝔽7), C11×ΓL(1,𝔽8)⋊D25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2310, 32);