Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1960 |
308 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(8,9,10,11)(13,18)(14,21)(19,23)(22,27);(2,3)(4,6,7,5)(12,14,15,22,16,21,24,27)(13,17,19,20,18,25,23,26);(4,5,7,6)(9,11)(12,13)(14,20)(15,23)(16,18)(17,22)(19,24)(21,26)(25,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=eg,d^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10682567, #10687581, #10690102, … |
| maximal normal subgroup of | #10682567, #10687581, #10690102, … |
| maximal quotient of | #10693976, #10694002, #10694182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2128);