Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1660 |
196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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)(4,5)(6,11)(7,9)(8,10)(12,15,17,13)(14,21,23,18)(16,19,26,24)(20,25,27,22);(6,10)(8,11)(12,14)(13,18)(15,21)(16,25)(17,23)(19,27)(20,24)(22,26);(2,3)(5,8)(6,10)(9,11)(12,13,17,15)(14,20,23,27)(16,24,26,19)(18,25,21,22)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,h^a=h^2,d^b=df,e^b=eg,d^c=dg,e^c=efg39 conjugacy classes — view the full character table.
| maximal subgroup of | #10715937, #10715938, D3×#1660, … |
| maximal normal subgroup of | #10715937, #10715938, D3×#1660, … |
| maximal quotient of | #10683979, #10683998, #10684015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4532);