Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2159 |
156 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 93 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 — 3 generators on 23 points:
(1,2,3)(9,15)(13,19)(16,20)(21,23);(8,9,14,21,11,16,18,19,12,15,22,23,17,20,10,13);(2,3)(4,5,6,7)(9,13)(10,14)(11,17)(15,19)(16,23)(18,22)(20,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=df,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,c^b=cg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10710607, #10710612, #10711709, … |
| maximal normal subgroup of | #10710607, #10710612, #10711709, … |
| maximal quotient of | #10706776, #10706840, #10706882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4004);