Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2165 |
128 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(5,10,9,12)(6,13)(11,19,16,18)(14,17);(1,2,3)(5,9)(6,13)(10,12)(14,17);(4,5,6,11,7,12,14,19,8,9,13,16,15,10,17,18)Polycyclic presentation:
2^7,3|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^a=cf,d^a=dg,e^b=eg,e^c=eg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10704616, #10704618, #10704620, … |
| maximal normal subgroup of | #10704616, #10704618, #10704620, … |
| maximal quotient of | #10710202, #10710255, #10710520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1690);