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ΓL(1,𝔽8)⋊D851 #49088383

order35742
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D851
unambiguous semidirect productC23⋊(ΓL(1,𝔽8)⋊D37)

Splittings

Fitting, socle, derived, HallC7×C23×C37 ⋊ C2×C3
HallΓL(1,𝔽8)×C23×C37 ⋊ C2
HallD5957 ⋊ C3
HallC23×C37 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8)×C37 ⋊ D23
HallΓL(1,𝔽8)×C23 ⋊ D37
HallC7×C37 ⋊ C3×D23
HallC7×C23 ⋊ C3×D37
SylowC37 ⋊ ΓL(1,𝔽8)⋊D23
SylowC23 ⋊ ΓL(1,𝔽8)⋊D37
HallΓL(1,𝔽8) ⋊ D851
SylowC7 ⋊ C3×D851

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C23×C37C2×C3
LCSG ▷ C7×C23×C37
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 5957

A minimal faithful permutation representation — 2 generators on 67 points:

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

Polycyclic presentation:

2,3,7,23,37|c^a=c^6,d^a=d^22,e^a=e^36,c^b=c^2

Conjugacy classes & character table

2132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35742, 46);