Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ C8×C4⋊C4 |
296 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(12,14,15,19,16,13,17,18);(1,2,3)(4,5,6,7)(9,11);(8,9,10,11)(12,13,15,18,16,14,17,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=de,e^2=g,b^a=bd,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×C4⋊C4, C2×#68651, #10793134, … |
| maximal normal subgroup of | D3×C8×C4⋊C4, C2×#68651, #10793134, … |
| maximal quotient of | #10686307, #10686315, #10686330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1227);