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C7×ΓL(1,𝔽8)×Dic17 #23756141

order9996
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)×Dic17
unambiguous semidirect product(C7×Dic17)⋊(C3×C7)

Splittings

derivedC7×C17 ⋊ C3×C4×C7
HallC72×Dic17 ⋊ C3
HallC7×C17×ΓL(1,𝔽8) ⋊ C4
HallC72×C17 ⋊ C3×C4
SylowC72 ⋊ C3×Dic17
SylowC17 ⋊ C4×C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2×C7ΓL(1,𝔽8)×D17
commutator, hypocenterC7×C17C3×C4×C7
FrattiniC2C7×ΓL(1,𝔽8)×D17
Fitting, socleC2×C72×C17C2×C3
prime radicalC2×C7×C17×ΓL(1,𝔽8)C2
cosocleC2×C7×C17C2×C3×C7
LCSG ▷ C7×C17
UCSC1 ◁ C2×C7

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

700 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)2×Dic17, C3×C7×ΓL(1,𝔽8)×Dic17, C7×ΓL(1,𝔽8)×Dic51
maximal normal subgroup ofΓL(1,𝔽8)2×Dic17, C3×C7×ΓL(1,𝔽8)×Dic17
maximal quotient ofC7×C7⋊C9×Dic17, C3×C7×ΓL(1,𝔽8)×Dic17, C7×ΓL(1,𝔽8)×Dic51

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 44);