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(C7⋊C8)⋊H(3,𝔽3) #13094150

order1512
automorphism?
homologyAbelianization (H1) = C32×C8 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect product(C7⋊C8)⋊H(3,𝔽3)
unambiguous semidirect product(C3×ΓL(1,𝔽8)⋊C8)⋊C3

Splittings

HallC7⋊H(3,𝔽3) ⋊ C8
HallC7⋊C8 ⋊ H(3,𝔽3)
SylowC7 ⋊ C8×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3×C4C3×AGL(1,𝔽7)
commutatorC3×C7C32×C8
FittingC32×C4×C7C2×C3
socleC2×C3×C7C32×C4
prime radicalC2×C7⋊H(3,𝔽3)C4
cosocleC3×C4×C7C2×C32
hypercenterC32×C4AGL(1,𝔽7)
hypocenterC7C8×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C32×C4

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 504

Polycyclic presentation:

2,3^2,2,3,2,7|a^2=d,d^2=f,g^a=g^6,c^b=ce,g^b=g^2

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 47);