Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#322 ⋊ C2 |
| Sylow | C3 ⋊ C8×M16 |
224 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(13,16)(17,19);(2,3)(4,5,6,8,7,9,10,11);(12,13,14,17,15,16,18,19)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cf,h^b=h^2,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×M16, C2×#68067, #10760768, … |
| maximal normal subgroup of | D3×C8×M16, C2×#68067, #10760768, … |
| maximal quotient of | #10684539, #10684567, #10685611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 643);