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H(3,𝔽3)×AGL(1,𝔽8) #13094591

order1512
automorphism?
homologyAbelianization (H1) = C32×C7 · Schur multiplier (H2) = C32

Aliases

direct productH(3,𝔽3)×AGL(1,𝔽8)
unambiguous semidirect productC32⋊(C3×AGL(1,𝔽8))
unambiguous semidirect product(C23×H(3,𝔽3))⋊C7

Splittings

Fitting, HallC23×H(3,𝔽3) ⋊ C7
SylowC23 ⋊ C7×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C32×AGL(1,𝔽8)
commutator, socle, cosocleC23×C3C32×C7
FittingC23×H(3,𝔽3)C7
hypercenterH(3,𝔽3)AGL(1,𝔽8)
hypocenterC23C7×H(3,𝔽3)
LCSG ▷ C23×C3 ▷ C23
UCSC1 ◁ C3 ◁ H(3,𝔽3)

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,4)(3,6,8)(5,9,7)(11,12,13,15,17,14,16);(2,5,6)(4,8,9)(10,14)(11,12)(13,17)(15,16)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 488);