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 ⋊ #1514 |
252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 252 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)(5,9)(8,10)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(5,8)(6,11)(9,10)(12,14)(13,17)(15,21)(16,22)(18,24)(19,20)(23,27)(25,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25)Polycyclic presentation:
2^7,3|a^2=e,c^2=fg,d^2=g,b^a=bd,c^a=cg,d^a=df,h^a=h^2,d^b=dg,e^b=efg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10687591, #10687592, #10687593, … |
| maximal normal subgroup of | #10687591, #10687592, #10687593, … |
| maximal quotient of | #10683272, #10683308, #10683311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4389);