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C2×SL(2,𝔽3)×S4 #11923478

order1152
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×SL(2,𝔽3)×S4
unambiguous semidirect productA4⋊(C22×SL(2,𝔽3))
unambiguous semidirect product(C2×A4)⋊(C2×SL(2,𝔽3))
unambiguous semidirect product(C2×Q8×S4)⋊C3

Splittings

FittingC23×Q8 ⋊ C3×D3
derivedQ8×A4 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC22A4×S4
commutator, cosocle, hypocenterQ8×A4C22×C3
FrattiniC2C2×A4×S4
FittingC23×Q8C3×D3
socleC24D3×A4
LCSG ▷ Q8×A4
UCSC1 ◁ C22

561 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 48

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×S4×GL(2,𝔽3), SL(2,𝔽3)×#1020⋊C2, C2×S4×PSU(3,𝔽2)⋊C3, …
maximal normal subgroup ofC2×S4×GL(2,𝔽3), SL(2,𝔽3)×GL(2,ℤ/4), #72244342
maximal quotient ofSL(2,𝔽3)×#1020⋊C2, C2×S4×PSU(3,𝔽2)⋊C3, SL(2,𝔽3)×GL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 157489);