Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1718 |
164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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)(6,13)(9,18)(12,19)(14,17)(20,22,24,30,23,29,31,35)(21,26,27,28,25,33,32,34);(4,6,7,14,8,13,16,17)(5,9,10,12,11,18,15,19)(21,25)(26,33)(27,32)(28,34);(2,3)(4,5,8,11)(6,12,13,19)(7,15,16,10)(9,17,18,14)(20,21)(22,28)(23,25)(24,32)(26,30)(27,31)(29,34)(33,35)Polycyclic presentation:
2^7,3|a^2=f,b^2=dg,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,c^b=cf66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1718, C2×#68096, #10782955, … |
| maximal normal subgroup of | D3×#1718, C2×#68096, #10782955, … |
| maximal quotient of | #10688982, #10694959, #10694973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 672);