Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #1878 |
338 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 113 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)(12,14,17,25)(13,19,21,27)(15,24,16,23)(18,22,20,26);(4,6,7,5)(9,11)(13,18)(15,16)(19,22)(20,21)(23,24)(26,27);(2,3)(4,5,7,6)(8,9,10,11)(12,13,16,18,17,21,15,20)(14,22,24,19,25,26,23,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,h^a=h^2,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1878, C2×#71632, #10796537, … |
| maximal normal subgroup of | D3×#1878, C2×#71632, #10796537, … |
| maximal quotient of | #10689755, #10689783, #10689927, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4208);