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C3×C16×SL(2,𝔽3) #11919376

order1152
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C32×C16 · Schur multiplier (H2) = C3

Aliases

direct productC3×C16×SL(2,𝔽3)
unambiguous semidirect product(Q8×C16)⋊C32
unambiguous semidirect product(C3×Q8×C16)⋊C3

Splittings

FittingC3×Q8×C16 ⋊ C3
derivedQ8 ⋊ C32×C16
SylowQ8×C16 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3×C16A4
commutator, hypocenterQ8C32×C16
FrattiniC2×C8C2×C3×A4
FittingC3×Q8×C16C3
socleC22×C3C8×A4
prime radicalC2×C3×SL(2,𝔽3)C8
cosocleC8×Q8C2×C32
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C16

146 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 384

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×SL(2,𝔽3)×C32, C3×C16×GL(2,𝔽3), C3×SL(2,𝔽3)⋊C32, …
maximal normal subgroup ofC3×SL(2,𝔽3)×C32, C3×C16×GL(2,𝔽3), C3×SL(2,𝔽3)⋊C32
maximal quotient ofC3×SL(2,𝔽3)×C32, C3×C16×PSU(3,𝔽2)⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 153387);