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 ⋊ #1969 |
312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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)(12,15,25,23,17,22,16,14)(13,19,26,18,21,27,20,24);(4,6,7,5)(9,11)(12,14,16,22,17,23,25,15)(13,18,20,19,21,24,26,27);(2,3)(4,5,7,6)(8,9,10,11)(12,13)(14,19)(15,24)(16,26)(17,21)(18,22)(20,25)(23,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=eg,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10681956, #10681958, #10712257, … |
| maximal normal subgroup of | #10681956, #10681958, #10712257, … |
| maximal quotient of | #10694045, #10694079, #10694343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2351);