Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1488 |
246 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,8)(9,11)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(2,3)(4,5,6,9,7,8,10,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,h^a=h^2,d^b=df78 conjugacy classes — view the full character table.
| maximal subgroup of | #10687039, #10687041, #10687046, … |
| maximal normal subgroup of | #10687039, #10687041, #10687046, … |
| maximal quotient of | #10683327, #10683340, #10683345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4216);