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C2×#10621518 #13062152

order1344
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×#10621518

Splittings

FittingC2×C7×#41 ⋊ C2×C3
HallC2×#6553 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×H(5,𝔽2)
SylowC7 ⋊ C2×C3×H(5,𝔽2)

Subgroups

subgroupG/H
centerC22C23×AGL(1,𝔽7)
commutator, cosocleC2×C7C25×C3
FrattiniC2C24×AGL(1,𝔽7)
FittingC2×C7×#41C2×C3
socleC22×C7C24×C3
hypercenterC2×#41AGL(1,𝔽7)
hypocenterC7C2×C3×H(5,𝔽2)
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×#41

1796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×H(5,𝔽2)×AGL(1,𝔽7), #72229268, #72229315, …
maximal normal subgroup ofC2×H(5,𝔽2)×AGL(1,𝔽7), #72229268, #72229315, …
maximal quotient of#72229268, #72229420, #72229433, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11441);