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C2×C13×ΓL(1,𝔽8)⋊D7 #21439723

order7644
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D7
automorphism?
homologyAbelianization (H1) = C22×C3×C13 · Schur multiplier (H2) = C2

Aliases

direct productC2×C13×ΓL(1,𝔽8)⋊D7
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C13×D7)

Splittings

Fitting, socleC2×C72×C13 ⋊ C2×C3
derived, SylowC72 ⋊ C22×C3×C13
HallC2×C13×C7⋊D7 ⋊ C3
HallC7×C13×ΓL(1,𝔽8) ⋊ C22
HallC72×C13 ⋊ C22×C3
HallC2×C7⋊D7 ⋊ C3×C13
HallC7×ΓL(1,𝔽8) ⋊ C22×C13

Subgroups

subgroupG/H
center, hypercenterC2×C13ΓL(1,𝔽8)⋊D7
commutator, hypocenterC72C22×C3×C13
FrattiniC1G
Fitting, socleC2×C72×C13C2×C3
LCSG ▷ C72
UCSC1 ◁ C2×C13

100 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

572 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C13⋊C3×ΓL(1,𝔽8)⋊D7, C2×C13×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C2×ΓL(1,𝔽8)⋊#10587721, …
maximal normal subgroup ofC2×C13⋊C3×ΓL(1,𝔽8)⋊D7, C2×C13×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C2×ΓL(1,𝔽8)⋊#10587721, …
maximal quotient ofC2×C13×(C7⋊C9)⋊D7, D3×C13×ΓL(1,𝔽8)⋊D7, C2×C3×C13×ΓL(1,𝔽8)⋊D7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7644, 292);