Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C8×SD16 |
180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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)(4,6,7,11,8,5,9,10);(2,3)(12,13,15,17)(14,19,18,16);(2,3)(4,5,7,10,8,6,9,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=e,d^2=f,e^2=g,b^a=bd,d^a=df,h^a=h^2,d^b=df,h^b=h^2,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×SD16, C2×#68149, C8×#912, … |
| maximal normal subgroup of | D3×C8×SD16, C2×#68149, C8×#912, … |
| maximal quotient of | #10685954, #10685957, #10686410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 725);