Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2056 |
288 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 85 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)(5,7)(9,11)(12,15,16,14,17,22,25,23)(13,19,20,24,21,27,26,18);(2,3)(8,9)(10,11)(12,14,17,23)(13,18,21,24)(15,25,22,16)(19,20,27,26);(2,3)(4,5)(6,7)(12,13,17,21)(14,19,23,27)(15,24,22,18)(16,26,25,20)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707646, #10707651, #10707657, … |
| maximal normal subgroup of | #10707646, #10707651, #10707657, … |
| maximal quotient of | #10699022, #10699029, #10699050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1440);