Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1863 |
298 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 298 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) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(12,14,17,18)(13,15,19,16);(2,3)(4,6,7,5)(9,11)(12,13)(14,15)(16,18)(17,19);(2,3)(4,5,7,6)(8,9)(10,11)(13,18)(14,19)(15,16)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=g,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×#1863, C2×#70236, #10796130, … |
| maximal normal subgroup of | D3×#1863, C2×#70236, #10796130, … |
| maximal quotient of | #10689107, #10689135, #10689244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2812);