Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C8×SL(2,𝔽3) |
| unambiguous semidirect product | (C2×C8×Q8)⋊C3 |
| Fitting, Sylow | C2×C8×Q8 ⋊ C3 |
| derived | Q8 ⋊ C2×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | A4 |
| commutator, hypocenter | Q8 | C2×C3×C8 |
| Frattini | C2×C4 | C22×A4 |
| Fitting | C2×C8×Q8 | C3 |
| socle | C23 | C4×A4 |
| prime radical | C22×SL(2,𝔽3) | C4 |
| cosocle | C4×Q8 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C8 | |
155 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 18 points:
(4,5,7)(8,10,9)(11,12,13,15,14,16,17,18);(1,2)(3,7,6,9)(4,10,8,5)112 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C8×SL(2,𝔽3), SL(2,𝔽3)×#97, #20569⋊C3, … |
| maximal normal subgroup of | C4×C8×SL(2,𝔽3), SL(2,𝔽3)×#97, #20569⋊C3, … |
| maximal quotient of | C4×C8×SL(2,𝔽3), SL(2,𝔽3)×C22⋊C8, SL(2,𝔽3)×C4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5825);