Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1846 |
248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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:
(1,2,3)(8,9,10,11)(13,18)(17,25)(19,23)(20,26);(12,14,15,21,16,22,24,27)(13,17,19,20,18,25,23,26);(2,3)(4,5,6,7)(9,11)(12,13)(14,20)(15,23)(16,18)(17,21)(19,24)(22,26)(25,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=d,c^2=eg,d^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10682366, #10682367, #10682376, … |
| maximal normal subgroup of | #10682366, #10682367, #10682376, … |
| maximal quotient of | #10688932, #10688947, #10688958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2342);