Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1387 |
136 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(8,9,10,14,18,20,23,21,12,16,19,13,11,15,17,22);(2,3)(4,5,6,7)(9,13)(10,17)(11,18)(14,16)(15,22)(19,23)(20,21)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=fg,e^2=fg,f^2=g,b^a=bc,c^a=cf,e^a=ef,f^a=fg,h^a=h^2,c^b=cg72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1387, C2×#67817, #10710112, … |
| maximal normal subgroup of | D3×#1387, C2×#67817, #10710112, … |
| maximal quotient of | #10682068, #10682117, #10682123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 393);