Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#405 ⋊ C2 |
| Sylow | C3 ⋊ #1931 |
232 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 77 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)(8,10,11,18,12,17,14,21)(9,15,13,19,16,23,20,22);(4,6,7,5)(8,9,14,20,12,16,11,13)(10,15,21,22,17,23,18,19);(2,3)(4,5,7,6)(9,13)(10,17)(11,14)(15,22)(16,20)(19,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=dfg,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^2,e^c=eg72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1931, C2×#69892, #10797730, … |
| maximal normal subgroup of | D3×#1931, C2×#69892, #10797730, … |
| maximal quotient of | #10693423, #10693437, #10693514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2468);