Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#490 ⋊ C2 |
| Sylow | C3 ⋊ #3142 |
392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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 — 4 generators on 27 points:
(1,2)(12,16,18,17)(13,21,23,22)(14,25,26,15)(19,24,27,20);(5,9)(8,11)(12,15,16,26,18,25,17,14)(13,20,22,19,23,24,21,27);(2,3)(4,6,7,10)(5,8,9,11)(12,14,17,25,18,26,16,15)(13,19,21,20,23,27,22,24);(4,5)(6,8)(7,9)(10,11)(12,13)(14,24)(15,19)(16,21)(17,22)(18,23)(20,26)(25,27)Polycyclic presentation:
2^7,3|b^2=f,c^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cfg,e^a=eg,f^a=fg,d^b=dg,h^b=h^2,d^c=dg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10706385, #10715539, #10776471, … |
| maximal normal subgroup of | #10706385, #10715539, #10776471, … |
| maximal quotient of | #10795406, #10797892, #10798861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14292);