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D12×H(5,𝔽2) #11707292

order768
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214

Aliases

direct productD12×H(5,𝔽2)

Splittings

FittingC3×C4×H(5,𝔽2) ⋊ C2
SylowC3 ⋊ D4×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC22C25×D3
commutator, socle, cosocleC22×C3C26
FittingC3×C4×H(5,𝔽2)C2
hypercenterC4×H(5,𝔽2)D3
hypocenterC3D4×H(5,𝔽2)
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×H(5,𝔽2)

12414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 96

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D4×H(5,𝔽2), D12×B(4,𝔽2), #72270988, …
maximal normal subgroup ofD3×D4×H(5,𝔽2), D12×B(4,𝔽2), #72270988, …
maximal quotient of#72247983, #72248031, #72248351, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1081673);