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(C2×AGL(1,ℤ/9))⋊C8 #11720439

order864
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product(C2×AGL(1,ℤ/9))⋊C8
unambiguous semidirect product((C2×D9)⋊C8)⋊C3

Splittings

FittingC2×C8×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C22⋊C8

Subgroups

subgroupG/H
centerC2×C4C2×AGL(1,ℤ/9)
commutatorC2×C9C2×C3×C8
FrattiniC2×C3×C4C2×C3×D3
FittingC2×C8×3−1+2C2
socleC22×C3C3×C4×D3
prime radicalC22×AGL(1,ℤ/9)C4
cosocleC2×C4×C9C22×C3
hypercenterC2×C8AGL(1,ℤ/9)
hypocenterC9C3×C22⋊C8
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C4 ◁ C2×C8

161 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 144

Polycyclic presentation:

2^2,3,2^3,3^2|b^2=d,d^2=f,g^3=h,b^a=be,g^a=g^2h^2,h^a=h^2,g^c=gh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10588367⋊C3, C8×C4⋊AGL(1,ℤ/9), #10588370⋊C3, …
maximal normal subgroup of#10588367⋊C3, C8×C4⋊AGL(1,ℤ/9), #10588370⋊C3, …
maximal quotient of#10588133⋊C3, #10588136⋊C3, #10588145⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 406);