Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1679 |
166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 166 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(7,11)(8,10);(2,3)(5,8)(9,10)(12,14,15,13,16,17,19,18);(4,5)(6,8)(7,9)(10,11)(12,13,16,18)(14,19,17,15)Polycyclic presentation:
2^7,3|a^2=f,b^2=dg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^b=h^2,e^c=eg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1679, C2×#71102, #10777494, … |
| maximal normal subgroup of | D3×#1679, C2×#71102, #10777494, … |
| maximal quotient of | #10688249, #10691338, #10691356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3678);