Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#343 ⋊ C2 |
| Sylow | C3 ⋊ #1822 |
272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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 — 3 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(13,18)(15,22)(17,23)(20,26);(2,3)(5,8)(9,11)(12,14,16,21)(13,17,18,23)(15,20,22,26)(19,24,25,27);(4,5)(6,9)(7,8)(10,11)(12,13,15,19,16,18,22,25)(14,17,20,24,21,23,26,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=d,f^2=g,b^a=bd,c^a=ce,e^a=eg,h^b=h^2,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10705875, #10705877, #10705885, … |
| maximal normal subgroup of | #10705875, #10705877, #10705885, … |
| maximal quotient of | #10688512, #10688515, #10688521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1567);