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C3×ΓL(1,𝔽8)×D31 #16840889

order3906
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×D31

Splittings

Fitting, socleC3×C7×C31 ⋊ C2×C3
derived, HallC7×C31 ⋊ C2×C32
HallC3×ΓL(1,𝔽8)×C31 ⋊ C2
HallC7×D31 ⋊ C32
SylowC31 ⋊ C2×C3×ΓL(1,𝔽8)
SylowC7 ⋊ C32×D31

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×D31
commutator, cosocle, hypocenterC7×C31C2×C32
FrattiniC1G
Fitting, socleC3×C7×C31C2×C3
LCSG ▷ C7×C31
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 651

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)×D31⋊C5, C3×C5×ΓL(1,𝔽8)×D31, C3×ΓL(1,𝔽8)×D155, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)×D31⋊C5, C3×C5×ΓL(1,𝔽8)×D31, C3×C7×ΓL(1,𝔽8)×D31
maximal quotient ofC3×C5×ΓL(1,𝔽8)×D31, C3×ΓL(1,𝔽8)×D155, C3×D31×C49⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3906, 39);