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C5⋊(ΓL(1,𝔽8)⋊D161) #46798948

order33810
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(ΓL(1,𝔽8)⋊D161)
unambiguous semidirect productΓL(1,𝔽8)⋊D805
unambiguous semidirect productC23⋊(C5⋊(ΓL(1,𝔽8)⋊D7))

Splittings

Fitting, socle, derived, HallC5×C72×C23 ⋊ C2×C3
HallC5×C7×ΓL(1,𝔽8)×C23 ⋊ C2
HallC5⋊(C7⋊D161) ⋊ C3
HallC7×ΓL(1,𝔽8)×C23 ⋊ D5
HallC72×C23 ⋊ C3×D5
HallC5×C7×ΓL(1,𝔽8) ⋊ D23
HallC5×C72 ⋊ C3×D23
HallC7×ΓL(1,𝔽8) ⋊ D115
HallC5×C23 ⋊ ΓL(1,𝔽8)⋊D7
SylowC72 ⋊ C3×D115
SylowC23 ⋊ C5⋊(ΓL(1,𝔽8)⋊D7)
SylowC5 ⋊ ΓL(1,𝔽8)⋊D161

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C23C2×C3
LCSG ▷ C5×C72×C23
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 2415

A minimal faithful permutation representation — 2 generators on 42 points:

(2,3)(4,5)(7,10,11,8,9,12)(14,15)(16,17)(18,19)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42);(1,2,4,5,3)(6,7,9,12,11,10,8)(13,14,16,18,19,17,15)(20,21,23,25,27,29,31,33,35,37,39,41,42,40,38,36,34,32,30,28,26,24,22)

Polycyclic presentation:

2,3,5,7^2,23|c^a=c^4,d^a=d^6,e^a=e^6,f^a=f^22,d^b=d^2

Conjugacy classes & character table

2017 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33810, 52);