Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1805 |
292 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(12,14,15,21)(13,18,17,26)(16,25,23,22)(19,27,24,20);(1,2,3)(8,9,10,11)(13,17)(18,26)(19,24)(20,27);(4,5,6,7)(9,11)(12,13)(14,20)(15,17)(16,24)(18,22)(19,23)(21,27)(25,26)Polycyclic presentation:
2^7,3|a^2=f,b^2=dg,c^2=g,e^2=g,b^a=bd,c^a=ce,e^a=eg,e^c=eg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10687159, #10687161, #10687173, … |
| maximal normal subgroup of | #10687159, #10687161, #10687173, … |
| maximal quotient of | #10687882, #10687902, #10687922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2135);