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C2×#11722641 #13164944

order1728
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×#11722641

Splittings

FittingC2×#41×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C2×H(5,𝔽2)

Subgroups

subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC2×C9C25×C3
FrattiniC2×C3C24×C3×D3
FittingC2×#41×3−1+2C2
socleC22×C3C23×C3×D3
hypercenterC2×#41AGL(1,ℤ/9)
hypocenterC9C2×C3×H(5,𝔽2)
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×#41

2776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×H(5,𝔽2)×AGL(1,ℤ/9), C2×#10588879⋊C3, #72244698, …
maximal normal subgroup ofC2×H(5,𝔽2)×AGL(1,ℤ/9), C2×#10588879⋊C3, #72244698, …
maximal quotient of#72244698, #72268843, #72268952, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 31007);