Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C4×#41⋊C3 |
| unambiguous semidirect product | (C2×C4×#41)⋊C3 |
| Fitting, Sylow | C2×C4×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C22 | C23×A4 |
| Fitting | C2×C4×#41 | C3 |
| socle | C23 | C22×A4 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | C2×Q8 | C23×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C42 | |
423 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(7,8,11,14)(9,13,17,20,10,19,16,22,15,12,18,21);(3,4,5,6)(7,15,11,17)(8,19,14,21)(9,18,16,10)(12,22,20,13);(1,2)112 conjugacy classes — view the full character table.
| maximal subgroup of | #20336⋊C3, #20343⋊C3, #22530⋊C3, … |
| maximal normal subgroup of | #20336⋊C3, #20343⋊C3, #22530⋊C3, … |
| maximal quotient of | C2×C42×SL(2,𝔽3), C42×#41⋊C3, C2×#2341⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18152);