Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #67⋊C16 |
| Fitting | C3×C22⋊C16 ⋊ C2 |
| Sylow | C3 ⋊ D4×C16 |
174 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3)(8,10,11,17,12,18,19,23,13,9,14,15,20,16,21,22);(1,2,3)(5,7);(4,5)(6,7)(8,9,11,15,12,16,19,22,13,10,14,17,20,18,21,23)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,e^2=f,f^2=g,b^a=bd,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4×C16, C2×#67⋊C16, #10948354, … |
| maximal normal subgroup of | D3×D4×C16, C2×#67⋊C16, #10948354, … |
| maximal quotient of | #10706889, #10706950, #10706956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1677);