Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #1864 |
362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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)(13,18)(14,15)(17,19);(2,3)(4,6,7,5)(9,11)(13,17)(14,15)(18,19);(2,3)(4,5,7,6)(8,9)(10,11)(12,13,16,19)(14,17,15,18)Polycyclic presentation:
2^7,3|a^2=f,b^2=fg,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^2,d^b=dg,e^b=eg,h^b=h^2,d^c=dg,e^c=eg66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1864, C2×#70170, #10796132, … |
| maximal normal subgroup of | D3×#1864, C2×#70170, #10796132, … |
| maximal quotient of | #10689114, #10689142, #10689265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2746);