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ΓL(1,𝔽8)⋊D767 #44567292

order32214
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D767
unambiguous semidirect productC59⋊(ΓL(1,𝔽8)⋊D13)

Splittings

Fitting, socle, derived, HallC7×C13×C59 ⋊ C2×C3
HallC13×ΓL(1,𝔽8)×C59 ⋊ C2
HallD5369 ⋊ C3
HallΓL(1,𝔽8)×C59 ⋊ D13
HallC13×C59 ⋊ AGL(1,𝔽7)
HallC7×C59 ⋊ C3×D13
HallC13×ΓL(1,𝔽8) ⋊ D59
HallC7×C13 ⋊ C3×D59
SylowC59 ⋊ ΓL(1,𝔽8)⋊D13
HallΓL(1,𝔽8) ⋊ D767
SylowC13 ⋊ ΓL(1,𝔽8)⋊D59
SylowC7 ⋊ C3×D767

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C13×C59C2×C3
LCSG ▷ C7×C13×C59
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 5369

A minimal faithful permutation representation — 2 generators on 79 points:

(2,5,6,3,4,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(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)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79);(1,2,4,7,6,5,3)(8,9,11,13,15,17,19,20,18,16,14,12,10)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23)

Polycyclic presentation:

2,3,7,13,59|c^a=c^6,d^a=d^12,e^a=e^58,c^b=c^2

Conjugacy classes & character table

1922 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32214, 46);