Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C3 ⋊ #2018 |
382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(9,15)(10,19)(11,12)(16,17)(18,22)(20,21);(5,7)(8,10)(9,14)(11,20)(12,19)(13,21)(15,22)(16,18)(17,23);(2,3)(4,5)(6,7)(8,9)(10,18)(11,16)(12,15)(13,17)(14,19)(20,23)(21,22)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eg,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2018, C2×#72095, #10822390, … |
| maximal normal subgroup of | D3×#2018, C2×#72095, #10822390, … |
| maximal quotient of | #10695784, #10695801, #10695852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4671);