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C2×#63512⋊C3 #13153729

order1728
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC2×#63512⋊C3
unambiguous semidirect product(C2×#63512)⋊C3

Splittings

FittingC2×#41×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C2×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC4×C9C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×#41×3−1+2C2
socleC22×C3C2×C3×#67
hypercenterC2×#41AGL(1,ℤ/9)
hypocenterC9C2×C3×AGL(1,ℤ/8)
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,ℤ/8)×AGL(1,ℤ/9), C2×#10588922⋊C3, C2×#10588502⋊C3, …
maximal normal subgroup ofC2×AGL(1,ℤ/8)×AGL(1,ℤ/9), C2×#10588922⋊C3, C2×#10588502⋊C3, …
maximal quotient of#72244722, #72252818

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19792);