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C5×ΓL(1,𝔽8)⋊D161 #46798928

order33810
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)⋊D161
commutator, cosocle, hypocenterC72×C23C2×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C23C2×C3
LCSG ▷ C72×C23
UCSC1 ◁ C5

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:

(1,2,3,4,5)(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);(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)

Conjugacy classes & character table

2035 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33810, 32);