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C2×ΓL(1,𝔽8)×C29⋊C7 #22555323

order8526
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC2×ΓL(1,𝔽8)×C29⋊C7
unambiguous semidirect product(C2×C7)⋊(C3×C29⋊C7)
unambiguous semidirect product(C2×C29)⋊(C7×ΓL(1,𝔽8))
unambiguous semidirect product(C2×ΓL(1,𝔽8)×C29)⋊C7
unambiguous semidirect product(C2×C7×C29⋊C7)⋊C3

Splittings

Fitting, socleC2×C7×C29 ⋊ C3×C7
derivedC7×C29 ⋊ C2×C3×C7
HallC2×C7×C29⋊C7 ⋊ C3
HallC7×C29⋊C7 ⋊ C2×C3
HallC2×C29 ⋊ C7×ΓL(1,𝔽8)
SylowC29 ⋊ C2×C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×C29⋊C7
commutator, hypocenterC7×C29C2×C3×C7
FrattiniC1G
Fitting, socleC2×C7×C29C3×C7
LCSG ▷ C7×C29
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊(Dic29⋊C7), ΓL(1,𝔽8)⋊C4×C29⋊C7, ΓL(1,𝔽8)×Dic29⋊C7, …
maximal normal subgroup ofΓL(1,𝔽8)⋊(Dic29⋊C7), ΓL(1,𝔽8)⋊C4×C29⋊C7, ΓL(1,𝔽8)×Dic29⋊C7, …
maximal quotient ofC4×ΓL(1,𝔽8)×C29⋊C7, C22×ΓL(1,𝔽8)×C29⋊C7, C2×C7⋊C9×C29⋊C7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8526, 21);