Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1485 |
218 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 85 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)(4,7,9,8)(5,10,6,11)(14,20)(17,23)(21,26)(24,27);(4,6,9,5)(7,11,8,10)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(4,5,8,11,9,6,7,10)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df,h^a=h^2,c^b=cfg,d^b=df66 conjugacy classes — view the full character table.
| maximal subgroup of | #10687047, #10687048, #10687061, … |
| maximal normal subgroup of | #10687047, #10687048, #10687061, … |
| maximal quotient of | #10683616, #10683639, #10683648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4401);