Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1837 |
188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,15,16,25,18,24,17,14)(13,20,21,27,23,26,22,19);(2,3)(4,6,7,10)(5,8,9,11)(12,14,17,24,18,25,16,15)(13,19,22,26,23,27,21,20);(4,5)(6,8)(7,9)(10,11)(12,13,16,22,18,23,17,21)(14,20,15,19,25,26,24,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10706148, #10706150, #10706153, … |
| maximal normal subgroup of | #10706148, #10706150, #10706153, … |
| maximal quotient of | #10688846, #10688849, #10688862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1504);