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ΓL(1,𝔽8)⋊D559 #37553535

order23478
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D559

Splittings

Fitting, socle, derived, HallC7×C13×C43 ⋊ C2×C3
HallC13×ΓL(1,𝔽8)×C43 ⋊ C2
HallD3913 ⋊ C3
HallΓL(1,𝔽8)×C43 ⋊ D13
HallC13×C43 ⋊ AGL(1,𝔽7)
HallC7×C43 ⋊ C3×D13
HallC13×ΓL(1,𝔽8) ⋊ D43
HallC7×C13 ⋊ C3×D43
SylowC43 ⋊ ΓL(1,𝔽8)⋊D13
HallΓL(1,𝔽8) ⋊ D559
SylowC13 ⋊ ΓL(1,𝔽8)⋊D43
SylowC7 ⋊ C3×D559

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C13×C43C2×C3
LCSG ▷ C7×C13×C43
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 63
transitive degreeμt(G) = 3913

A minimal faithful permutation representation — 2 generators on 63 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)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(58,61,62,59,60,63);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,45,47,49,51,53,55,56,54,52,50,48,46)(57,58,60,63,62,61,59)

Polycyclic presentation:

2,3,7,13,43|c^a=c^6,d^a=d^12,e^a=e^42,c^b=c^2

Conjugacy classes & character table

1402 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23478, 124);