Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2205 |
224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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:
(1,4,7,13)(2,9,12,10)(3,5,6,15)(8,16,14,11)(17,18,19);(1,3,7,6)(2,8,12,14)(4,15,13,5)(9,11,10,16)(18,19);(1,2)(3,11)(4,14)(5,9)(6,16)(7,12)(8,13)(10,15)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cg,d^b=dg,e^b=efg,h^b=h^2,d^c=df,e^c=eg,e^d=eg36 conjugacy classes — view the full character table.
| maximal subgroup of | #10715956, #10715957, #10715961, … |
| maximal normal subgroup of | #10715956, #10715957, #10715961, … |
| maximal quotient of | #10707277, #10707278, #10707442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3571);