Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2078 |
198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 198 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(4,6,7,10)(5,9,8,11)(13,19)(14,22)(15,16)(18,25)(20,21)(23,24);(1,2,3)(5,8)(9,11)(12,14)(13,18)(15,23)(16,22)(17,24)(19,27)(20,26)(21,25);(4,5)(6,9)(7,8)(10,11)(12,13,15,20,17,21,16,19)(14,18,23,26,24,25,22,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,c^b=cf,f^b=fg,e^c=eg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10708423, #10708425, #10708430, … |
| maximal normal subgroup of | #10708423, #10708425, #10708430, … |
| maximal quotient of | #10699887, #10699892, #10699897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2570);