Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1319 |
134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(2,3)(4,5,6,8,7,9,10,11)(12,13)(14,20,22,27)(15,23,25,18)(16,19)(17,21)(24,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=g,c^2=g,d^2=f,b^a=bc,c^a=ce,e^a=eg,h^a=h^2,c^b=cg,d^b=deg,d^c=dg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682549, #10682550, #10682616, … |
| maximal normal subgroup of | #10682549, #10682550, #10682616, … |
| maximal quotient of | #10681812, #10681820, #10681935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 219);