Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2175 |
160 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(4,6,7,5)(9,14)(13,21)(15,22)(16,20);(9,13)(10,17)(11,18)(14,21)(15,16)(19,23)(20,22);(2,3)(4,5,7,6)(8,9)(10,13)(11,20)(12,14)(15,17)(16,18)(19,21)(22,23)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^2,d^b=df,f^b=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2175, C2×#71474, #10948492, … |
| maximal normal subgroup of | D3×#2175, C2×#71474, #10948492, … |
| maximal quotient of | #10706927, #10707118, #10707125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4050);