Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2224 |
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(13,17)(14,21)(15,22)(18,19)(20,26)(23,27)(24,25);(2,3)(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,24)(16,20)(18,21)(19,22)(23,26)(25,27)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^b=h^2,e^c=ef,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2224, C2×#69178, #10949829, … |
| maximal normal subgroup of | D3×#2224, C2×#69178, #10949829, … |
| maximal quotient of | #10711226, #10711777, #10711781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1754);