Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×#101 ⋊ C2 |
| Sylow | C3 ⋊ #1888 |
344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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)(8,11,13,20)(9,15,10,18)(12,21,14,19)(16,23,17,22);(2,3)(4,6,7,5)(8,10,13,9)(11,18,20,15)(12,16,14,17)(19,22,21,23);(4,5,7,6)(8,9,14,17,13,10,12,16)(11,15,21,23,20,18,19,22)Polycyclic presentation:
2^7,3|a^2=f,b^2=dfg,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eg,f^b=fg,h^b=h^2,d^c=dg,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10711062, #10711065, #10711070, … |
| maximal normal subgroup of | #10711062, #10711065, #10711070, … |
| maximal quotient of | #10689845, #10689865, #10690119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2680);