Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1293 |
184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 184 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) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,6)(5,8)(7,11)(9,10)(12,14,16,24,17,25,15,23)(13,18,20,26,21,27,19,22);(2,3)(4,5)(6,10,11,8)(7,9)(12,13,15,20,17,21,16,19)(14,22,23,18,25,26,24,27)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=fg,d^2=fg,e^2=fg,b^a=bc,c^a=cf,e^a=efg,h^a=h^2,d^b=dg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10681884, #10681885, #10681887, … |
| maximal normal subgroup of | #10681884, #10681885, #10681887, … |
| maximal quotient of | #10681725, #10681727, #10681790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 255);