Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1608 |
228 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 228 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) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,15)(13,18)(14,20)(16,22)(17,23)(19,25)(21,27)(24,26);(2,3)(4,5,7,9)(6,11,10,8)(12,14)(13,17)(15,20)(16,21)(18,23)(19,24)(22,27)(25,26);(2,3)(5,8)(6,10)(9,11)(12,13,16,19)(14,17,21,24)(15,18,22,25)(20,26,27,23)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10707910, #10707911, #10707916, … |
| maximal normal subgroup of | #10707910, #10707911, #10707916, … |
| maximal quotient of | #10684218, #10684246, #10684314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 692);