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C29×C11⋊AGL(1,𝔽7) #27211540

order13398
isoclinismisoclinic class of C11⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C29 · Schur multiplier (H2) = C1

Aliases

direct productC29×C11⋊AGL(1,𝔽7)
unambiguous semidirect productC11⋊(C29×AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(D11×C29)
unambiguous semidirect productD77⋊(C3×C29)
unambiguous semidirect product(C29×D77)⋊C3

Splittings

Fitting, socle, HallC7×C11×C29 ⋊ C2×C3
derived, HallC7×C11 ⋊ C2×C3×C29
HallC11×ΓL(1,𝔽8)×C29 ⋊ C2
HallC29×D77 ⋊ C3
HallΓL(1,𝔽8)×C29 ⋊ D11
HallC11×C29 ⋊ AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C2×C29
HallC7×C29 ⋊ C3×D11
HallD77 ⋊ C3×C29
HallΓL(1,𝔽8) ⋊ D11×C29
SylowC11 ⋊ C29×AGL(1,𝔽7)
SylowC7 ⋊ C3×D11×C29

Subgroups

subgroupG/H
center, hypercenterC29C11⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC7×C11C2×C3×C29
FrattiniC1G
Fitting, socleC7×C11×C29C2×C3
LCSG ▷ C7×C11
UCSC1 ◁ C29

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 2233

A minimal faithful permutation representation — 2 generators on 47 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(13,16,17,14,15,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(1,2,4,6,8,10,11,9,7,5,3)(12,13,15,18,17,16,14)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofD11×C29×AGL(1,𝔽7), (C11⋊AGL(1,𝔽7))⋊D29, D29×C11⋊AGL(1,𝔽7), …
maximal normal subgroup ofD11×C29×AGL(1,𝔽7), (C11⋊AGL(1,𝔽7))⋊D29, D29×C11⋊AGL(1,𝔽7), …
maximal quotient ofC29×C11⋊(ΓL(1,𝔽8)⋊C4), C2×C29×C11⋊AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13398, 31);