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ΓL(1,𝔽8)×D319 #27211537

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

Aliases

direct productΓL(1,𝔽8)×D319
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D29)
unambiguous semidirect productC29⋊(ΓL(1,𝔽8)×D11)
unambiguous semidirect product(C7×D319)⋊C3

Splittings

Fitting, socle, derived, HallC7×C11×C29 ⋊ C2×C3
HallC11×ΓL(1,𝔽8)×C29 ⋊ C2
HallC7×D319 ⋊ C3
HallΓL(1,𝔽8)×C29 ⋊ D11
HallC11×C29 ⋊ C2×ΓL(1,𝔽8)
HallC11×ΓL(1,𝔽8) ⋊ D29
HallC7×C29 ⋊ C3×D11
HallC7×C11 ⋊ C3×D29
SylowC29 ⋊ ΓL(1,𝔽8)×D11
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)(42,43,44)(45,47,46);(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)(41,42,43,45,44,46,47)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,𝔽7)×D319, (C11⋊AGL(1,𝔽7))⋊D29, ΓL(1,𝔽8)×D11×D29, …
maximal normal subgroup ofAGL(1,𝔽7)×D319, (C11⋊AGL(1,𝔽7))⋊D29, ΓL(1,𝔽8)×D11×D29, …
maximal quotient ofΓL(1,𝔽8)×Dic319, C2×ΓL(1,𝔽8)×D319

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13398, 28);