Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4⋊C4×SL(2,𝔽3) |
| unambiguous semidirect product | C4⋊(C4×SL(2,𝔽3)) |
| unambiguous semidirect product | (Q8×C4⋊C4)⋊C3 |
| Fitting, Sylow | Q8×C4⋊C4 ⋊ C3 |
193 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 193 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) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 16 points:
(5,6,7,8)(10,11,13)(14,16,15);(1,2,3,4)(6,8)(9,13,12,15)(10,16,14,11)70 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×#102, SL(2,𝔽3)×#103, #22540⋊C3, … |
| maximal normal subgroup of | SL(2,𝔽3)×#102, SL(2,𝔽3)×#103, #22540⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#95, SL(2,𝔽3)×C4⋊C8, SL(2,𝔽3)×#106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5752);