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ΓL(1,𝔽8)×D11×C29 #27211536

order13398
automorphism?
homologyAbelianization (H1) = C2×C3×C29 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D11×C29
unambiguous semidirect productC11⋊(C2×ΓL(1,𝔽8)×C29)
unambiguous semidirect product(C7×D11)⋊(C3×C29)
unambiguous semidirect product(C7×D11×C29)⋊C3

Splittings

Fitting, socle, HallC7×C11×C29 ⋊ C2×C3
derived, HallC7×C11 ⋊ C2×C3×C29
HallC11×ΓL(1,𝔽8)×C29 ⋊ C2
HallC7×D11×C29 ⋊ C3
HallC11×C29 ⋊ C2×ΓL(1,𝔽8)
HallC11×ΓL(1,𝔽8) ⋊ C2×C29
HallC7×C29 ⋊ C3×D11
HallC7×D11 ⋊ C3×C29
SylowC11 ⋊ C2×ΓL(1,𝔽8)×C29
SylowC7 ⋊ C3×D11×C29

Subgroups

subgroupG/H
center, hypercenterC29ΓL(1,𝔽8)×D11
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)(13,14,15)(16,18,17)(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)(12,13,14,16,15,17,18)

Conjugacy classes & character table

1015 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD11×C29×AGL(1,𝔽7), D11×ΓL(1,𝔽8)⋊D29, ΓL(1,𝔽8)×D11×D29, …
maximal normal subgroup ofD11×C29×AGL(1,𝔽7), D11×ΓL(1,𝔽8)⋊D29, ΓL(1,𝔽8)×D11×D29, …
maximal quotient ofΓL(1,𝔽8)×C29×Dic11, C2×ΓL(1,𝔽8)×D11×C29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13398, 27);