Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68629 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2099 |
214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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)(5,9)(8,11)(12,14,15,19)(13,17,16,18)(21,24)(22,27)(25,26);(2,3)(4,6,7,10)(5,8,9,11)(13,16)(14,19)(20,21)(22,24)(23,26)(25,27);(4,5)(6,8)(7,9)(10,11)(12,13,15,16)(14,18,19,17)Polycyclic presentation:
2^7,3|a^2=d,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cf,f^b=fg,h^b=h^2,f^c=fg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10684217, #10684218, #10684224, … |
| maximal normal subgroup of | #10684217, #10684218, #10684224, … |
| maximal quotient of | #10700313, #10703098, #10703446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1218);