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C11⋊(ΓL(1,𝔽8)⋊D59) #41030673

order27258
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC11⋊(ΓL(1,𝔽8)⋊D59)
unambiguous semidirect productΓL(1,𝔽8)⋊D649
unambiguous semidirect productC59⋊(C11⋊AGL(1,𝔽7))
unambiguous semidirect productD4543⋊C3

Splittings

Fitting, socle, derived, HallC7×C11×C59 ⋊ C2×C3
HallC11×ΓL(1,𝔽8)×C59 ⋊ C2
HallD4543 ⋊ C3
HallΓL(1,𝔽8)×C59 ⋊ D11
HallC11×C59 ⋊ AGL(1,𝔽7)
HallC7×C59 ⋊ C3×D11
HallC11×ΓL(1,𝔽8) ⋊ D59
HallC7×C11 ⋊ C3×D59
SylowC59 ⋊ C11⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D649
SylowC11 ⋊ ΓL(1,𝔽8)⋊D59
SylowC7 ⋊ C3×D649

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C59C2×C3
LCSG ▷ C7×C11×C59
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 77
transitive degreeμt(G) = 4543

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

Polycyclic presentation:

2,3,7,11,59|c^a=c^6,d^a=d^10,e^a=e^58,c^b=c^2

Conjugacy classes & character table

1627 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27258, 24);