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C2×C4×(C22⋊D7)⋊C3 #13057522

order1344
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×C4×(C22⋊D7)⋊C3
unambiguous semidirect product(C2×C4×C22⋊D7)⋊C3

Splittings

FittingC23×C4×C7 ⋊ C2×C3
HallC2×C4×C22⋊D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4×D4
SylowC7 ⋊ C2×C3×C4×D4

Subgroups

subgroupG/H
centerC22×C4C2×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC23×C4×C7C2×C3
socleC23×C7C23×C3
prime radicalC22×(C22⋊D7)⋊C3C2
cosocleC22×C7C24×C3
hypercenterC23×C4AGL(1,𝔽7)
hypocenterC7C2×C3×C4×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C23×C4

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽5)×(C22⋊D7)⋊C3, C2×C4×D4×AGL(1,𝔽7), C4×#10620789, …
maximal normal subgroup ofC2×C4×D4×AGL(1,𝔽7), C4×#10620789, C4×#6445⋊C3, …
maximal quotient ofC2×AGL(1,𝔽5)×(C22⋊D7)⋊C3, C4×#6495⋊C3, C4×#10620789, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6811);