Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1636 |
188 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 55 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)(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)(20,23,26,22)(21,27,30,35)(24,32,25,33)(28,34,29,31);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16)(20,22,26,23)(21,27,30,35)(24,33,25,32)(28,34,29,31);(2,3)(4,5,9,12)(6,13,15,18)(7,11,8,16)(10,17,19,14)(20,21,26,30)(22,31,23,34)(24,29,25,28)(27,33,35,32)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,d^c=dg,e^c=eg,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1636, C2×#68458, #10770944, … |
| maximal normal subgroup of | D3×#1636, C2×#68458, #10770944, … |
| maximal quotient of | #10684051, #10684093, #10684107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1034);