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SL(2,𝔽3)×C32 #11709194

order768
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C3×C32 · Schur multiplier (H2) = C1

Aliases

direct productSL(2,𝔽3)×C32
unambiguous semidirect product(Q8×C32)⋊C3

Splittings

Fitting, SylowQ8×C32 ⋊ C3
derivedQ8 ⋊ C3×C32

Subgroups

subgroupG/H
center, hypercenterC2×C32A4
commutator, hypocenterQ8C3×C32
FrattiniC2×C16C2×A4
FittingQ8×C32C3
socleC22A4×C16
prime radicalC2×SL(2,𝔽3)C16
cosocleQ8×C16C2×C3
LCSG ▷ Q8
UCSC1 ◁ C2×C32

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 256

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×SL(2,𝔽3)×C32, SL(2,𝔽3)×C64, C32×GL(2,𝔽3), …
maximal normal subgroup ofC3×SL(2,𝔽3)×C32, SL(2,𝔽3)×C64, C32×GL(2,𝔽3)
maximal quotient ofC3×SL(2,𝔽3)×C32, SL(2,𝔽3)×C64, C32×PSU(3,𝔽2)⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1083575);