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H(3,𝔽3)×D29 #13095809

order1566
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

direct productH(3,𝔽3)×D29
unambiguous semidirect productC29⋊(C2×H(3,𝔽3))
unambiguous semidirect product(C32×D29)⋊C3

Splittings

Fitting, HallH(3,𝔽3)×C29 ⋊ C2
SylowC29 ⋊ C2×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C32×D29
commutator, socle, cosocleC3×C29C2×C32
FittingH(3,𝔽3)×C29C2
hypercenterH(3,𝔽3)D29
hypocenterC29C2×H(3,𝔽3)
LCSG ▷ C3×C29 ▷ C29
UCSC1 ◁ C3 ◁ H(3,𝔽3)

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 261

A minimal faithful permutation representation — 2 generators on 38 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1566, 6);