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ΓL(1,𝔽8)⋊D551 #37529046

order23142
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D551
unambiguous semidirect productC29⋊(ΓL(1,𝔽8)⋊D19)

Splittings

Fitting, socle, derived, HallC7×C19×C29 ⋊ C2×C3
HallC19×ΓL(1,𝔽8)×C29 ⋊ C2
HallD3857 ⋊ C3
HallΓL(1,𝔽8)×C29 ⋊ D19
HallC19×C29 ⋊ AGL(1,𝔽7)
HallC19×ΓL(1,𝔽8) ⋊ D29
HallC7×C29 ⋊ C3×D19
HallC7×C19 ⋊ C3×D29
SylowC29 ⋊ ΓL(1,𝔽8)⋊D19
HallΓL(1,𝔽8) ⋊ D551
SylowC19 ⋊ ΓL(1,𝔽8)⋊D29
SylowC7 ⋊ C3×D551

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C19×C29C2×C3
LCSG ▷ C7×C19×C29
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 3857

A minimal faithful permutation representation — 2 generators on 55 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)(41,42)(43,44)(45,46)(47,48)(50,53,54,51,52,55);(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)(30,31,33,35,37,39,41,43,45,47,48,46,44,42,40,38,36,34,32)(49,50,52,55,54,53,51)

Polycyclic presentation:

2,3,7,19,29|c^a=c^6,d^a=d^18,e^a=e^28,c^b=c^2

Conjugacy classes & character table

1382 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23142, 55);