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C11⋊(ΓL(1,𝔽8)⋊D29) #27211541

order13398
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC11⋊(ΓL(1,𝔽8)⋊D29)
unambiguous semidirect productΓL(1,𝔽8)⋊D319
unambiguous semidirect productC29⋊(C11⋊AGL(1,𝔽7))
unambiguous semidirect productD2233⋊C3

Splittings

Fitting, socle, derived, HallC7×C11×C29 ⋊ C2×C3
HallC11×ΓL(1,𝔽8)×C29 ⋊ C2
HallD2233 ⋊ C3
HallΓL(1,𝔽8)×C29 ⋊ D11
HallC11×C29 ⋊ AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ D29
HallC7×C29 ⋊ C3×D11
HallC7×C11 ⋊ C3×D29
SylowC29 ⋊ C11⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D319
SylowC11 ⋊ ΓL(1,𝔽8)⋊D29
SylowC7 ⋊ C3×D319

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C29C2×C3
LCSG ▷ C7×C11×C29
UCSC1

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

Polycyclic presentation:

2,3,7,11,29|c^a=c^6,d^a=d^10,e^a=e^28,c^b=c^2

Conjugacy classes & character table

802 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×ΓL(1,𝔽8)⋊D29, AGL(1,𝔽7)×D319, D29×C11⋊AGL(1,𝔽7), …
maximal normal subgroup ofD11×ΓL(1,𝔽8)⋊D29, AGL(1,𝔽7)×D319, D29×C11⋊AGL(1,𝔽7), …
maximal quotient ofC11⋊(ΓL(1,𝔽8)⋊Dic29), C2×C11⋊(ΓL(1,𝔽8)⋊D29)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13398, 32);