Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#451 ⋊ C2 |
| Sylow | C3 ⋊ #3108 |
390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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)(4,6,7,10)(5,9,8,11);(12,14,15,21)(13,18,17,26)(16,25,23,22)(19,27,24,20);(5,8)(9,11)(13,17)(18,26)(19,24)(20,27);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,20)(15,17)(16,24)(18,22)(19,23)(21,27)(25,26)Polycyclic presentation:
2^7,3|c^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,f^c=fg,h^d=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10767523, #10767525, #10767543, … |
| maximal normal subgroup of | #10767523, #10767525, #10767543, … |
| maximal quotient of | #10795201, #10798145, #10798185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12749);