Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #3152 |
552 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,17)(13,22)(16,23)(18,21);(4,5,6,7);(2,3)(8,10,11,18,12,17,20,21)(9,13,14,16,15,22,19,23);(2,3)(5,7)(8,9,12,15)(10,16,17,23)(11,19,20,14)(13,21,22,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,d^b=dg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10692702, #10692704, #10692743, … |
| maximal normal subgroup of | #10692702, #10692704, #10692743, … |
| maximal quotient of | #10783043, #10783087, #10783126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15864);