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C17⋊(ΓL(1,𝔽8)⋊D53) #49180928

order37842
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC17⋊(ΓL(1,𝔽8)⋊D53)
unambiguous semidirect productΓL(1,𝔽8)⋊D901
unambiguous semidirect productC53⋊(C17⋊AGL(1,𝔽7))
unambiguous semidirect productD6307⋊C3

Splittings

Fitting, socle, derived, HallC7×C17×C53 ⋊ C2×C3
HallC17×ΓL(1,𝔽8)×C53 ⋊ C2
HallD6307 ⋊ C3
HallΓL(1,𝔽8)×C53 ⋊ D17
HallC17×C53 ⋊ AGL(1,𝔽7)
HallC7×C53 ⋊ C3×D17
HallC17×ΓL(1,𝔽8) ⋊ D53
HallC7×C17 ⋊ C3×D53
SylowC53 ⋊ C17⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D901
SylowC17 ⋊ ΓL(1,𝔽8)⋊D53
SylowC7 ⋊ C3×D901

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C17×C53C2×C3
LCSG ▷ C7×C17×C53
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 77
transitive degreeμt(G) = 6307

A minimal faithful permutation representation — 2 generators on 77 points:

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

Polycyclic presentation:

2,3,7,17,53|c^a=c^6,d^a=d^16,e^a=e^52,c^b=c^2

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37842, 24);