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C7×(C22⋊D7)⋊C3 #11924192

order1176
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×(C22⋊D7)⋊C3
unambiguous semidirect product(C22⋊D7)⋊(C3×C7)

Splittings

FittingC22×C72 ⋊ C2×C3
HallC7×C22⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ D4
SylowC72 ⋊ C3×D4

Subgroups

subgroupG/H
centerC2×C7C2×AGL(1,𝔽7)
commutator, cosocleC2×C7C22×C3×C7
FrattiniC2C2×C7×AGL(1,𝔽7)
FittingC22×C72C2×C3
socleC2×C72C22×C3
hypercenterC22×C7AGL(1,𝔽7)
hypocenterC7C3×C7×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C7 ◁ C22×C7

69 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 84

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×(C22⋊D7)⋊C3, #72222021, C7×#1157⋊C3, …
maximal normal subgroup ofΓL(1,𝔽8)×(C22⋊D7)⋊C3, #72222021, C7×#1157⋊C3, …
maximal quotient ofC7×#1142⋊C3, C7×#1141⋊C3, C2×C7×(C22⋊D7)⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 107);