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Q8⋊AGL(1,𝔽9) #10593270

order576
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productQ8⋊AGL(1,𝔽9)

Splittings

FittingC32×Q8 ⋊ C8
SylowC32 ⋊ Q8⋊C8

Subgroups

subgroupG/H
centerC2C22⋊AGL(1,𝔽9)
commutatorC32×C4C2×C8
FrattiniC4C2×AGL(1,𝔽9)
FittingC32×Q8C8
socleC2×C32C22⋊C8
prime radicalC2×C3⋊D3C22⋊C4
cosocleC4×#157C22
hypercenterQ8AGL(1,𝔽9)
hypocenterC32Q8⋊C8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C4 ◁ Q8

89 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Polycyclic presentation:

2^6,3^2|a^2=c,b^2=f,c^2=e,d^2=f,b^a=bd,d^a=df,g^a=g^2h^2,h^a=g^2,d^b=df,g^c=g^2h,h^c=gh,g^e=g^2,h^e=h^2

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×Q8⋊AGL(1,𝔽9), #11774254, #11774291, …
maximal normal subgroup ofC2×Q8⋊AGL(1,𝔽9), #11774254, #11774291, …
maximal quotient of#11770678, #11770693, #11770796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 5152);