Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1520 |
288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(6,11)(8,10)(12,14,16,15)(13,18,19,17);(2,3)(4,6,7,11)(5,8,9,10)(12,13,16,19)(14,17,15,18);(2,3)(4,5)(6,10,11,8)(7,9)(13,17)(14,15)(18,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=g,d^2=g,b^a=bd,c^a=cg,d^a=df,h^a=h^2,c^b=cf,d^b=dg,e^b=efg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10686721, #10686722, #10686726, … |
| maximal normal subgroup of | #10686721, #10686722, #10686726, … |
| maximal quotient of | #10683184, #10683188, #10683250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2522);