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SL(2,𝔽3)×AGL(1,ℤ/9) #13048437

order1296
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productSL(2,𝔽3)×AGL(1,ℤ/9)
unambiguous semidirect product(D9⋊SL(2,𝔽3))⋊C3
unambiguous semidirect product(D9×SL(2,𝔽3))⋊C3
unambiguous semidirect product(Q8×AGL(1,ℤ/9))⋊C3

Splittings

FittingQ8×3−1+2 ⋊ C2×C3
derivedQ8×C9 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC2A4×AGL(1,ℤ/9)
commutator, cosocle, hypocenterQ8×C9C2×C32
Frattini, socleC2×C3C3×D3×A4
FittingQ8×3−1+2C2×C3
LCSG ▷ Q8×C9
UCSC1 ◁ C2

181 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/9)×#997, GL(2,𝔽3)×AGL(1,ℤ/9), C2×SL(2,𝔽3)×AGL(1,ℤ/9), …
maximal normal subgroup ofGL(2,𝔽3)×AGL(1,ℤ/9), C2×SL(2,𝔽3)×AGL(1,ℤ/9)
maximal quotient ofC2×SL(2,𝔽3)×AGL(1,ℤ/9), #72253007

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2155);