Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1663 |
172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 172 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) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,6)(5,9)(7,16)(8,13)(10,19)(11,18)(12,14)(15,17)(20,23,25,21)(22,29,31,26)(24,27,34,32)(28,33,35,30);(4,5,8,11)(6,9,13,18)(7,15,14,10)(12,19,16,17)(20,22,25,31)(21,26,23,29)(24,33,34,30)(27,35,32,28);(2,3)(6,12)(7,14)(9,17)(10,15)(13,16)(18,19)(20,21,25,23)(22,28,31,35)(24,32,34,27)(26,33,29,30)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,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 | D3×#1663, C2×#71961, #10776382, … |
| maximal normal subgroup of | D3×#1663, C2×#71961, #10776382, … |
| maximal quotient of | #10684050, #10684085, #10684113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4537);