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C2×ΓL(1,𝔽8)⋊D91 #21439728

order7644
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)⋊D91
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D91

Splittings

Fitting, socleC2×C72×C13 ⋊ C2×C3
derived, HallC72×C13 ⋊ C22×C3
HallC2×C7⋊D91 ⋊ C3
HallC7×C13×ΓL(1,𝔽8) ⋊ C22
HallC7×ΓL(1,𝔽8) ⋊ C2×D13
SylowC72 ⋊ C2×C3×D13
SylowC13 ⋊ C2×ΓL(1,𝔽8)⋊D7

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)⋊D91
commutator, hypocenterC72×C13C22×C3
FrattiniC1G
Fitting, socleC2×C72×C13C2×C3
LCSG ▷ C72×C13
UCSC1 ◁ C2

100 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

464 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×ΓL(1,𝔽8)⋊((C13⋊C3)⋊D7), C2×ΓL(1,𝔽8)⋊#10587722, C2×ΓL(1,𝔽8)⋊#10587723, …
maximal normal subgroup ofC2×ΓL(1,𝔽8)⋊((C13⋊C3)⋊D7), C2×ΓL(1,𝔽8)⋊#10587722, C2×ΓL(1,𝔽8)⋊#10587723, …
maximal quotient ofC2×(C7⋊C9)⋊D91, D3×ΓL(1,𝔽8)⋊D91, C2×C3×ΓL(1,𝔽8)⋊D91, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7644, 297);