Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1988 |
292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,11,15)(9,12,13,14);(2,3)(4,6,7,5)(17,20)(18,23)(21,22);(2,3)(4,5,7,6)(8,9,11,13)(10,14,15,12)(16,17,18,21,19,22,23,20)Polycyclic presentation:
2^7,3|a^2=f,b^2=deg,c^2=efg,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=cefg,h^a=h^2,d^b=dg,e^b=eg,f^b=fg,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10687103, #10687104, #10687145, … |
| maximal normal subgroup of | #10687103, #10687104, #10687145, … |
| maximal quotient of | #10695173, #10695191, #10695313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4356);