Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2255 |
120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,13,19,25,15,20,22,23,16,18,26,27,21,24,14,17);(2,3)(4,6,7,11)(5,8,9,10)(13,18)(17,23)(20,24)(25,27);(4,5,7,9)(6,10,11,8)(13,17)(14,19)(15,21)(18,23)(20,27)(22,26)(24,25)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,c^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,c^b=cg,h^b=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2255, C2×#69322, #10950686, … |
| maximal normal subgroup of | D3×#2255, C2×#69322, #10950686, … |
| maximal quotient of | #10711629, #10713225, #10713226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1898);