Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1776 |
272 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 129 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:
(1,2,3)(8,10,11,9)(13,18)(17,19);(4,5,6,7)(13,17)(14,15)(18,19);(2,3)(8,9,11,10)(12,13,15,17,16,18,14,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,b^a=bd,c^a=cg,h^a=h^2,d^b=dg,e^b=eg84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1776, C2×#69481, #10793246, … |
| maximal normal subgroup of | D3×#1776, C2×#69481, #10793246, … |
| maximal quotient of | #10686385, #10686390, #10686406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2057);