Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1591 |
152 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 69 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)(4,6,8,9,7,10,11,5);(2,3)(5,9)(6,10)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(2,3)(4,5,8,6,7,9,11,10)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,h^a=h^2,c^b=cg,d^b=df,h^b=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10681850, #10681852, #10681856, … |
| maximal normal subgroup of | #10681850, #10681852, #10681856, … |
| maximal quotient of | #10686014, #10686421, #10689136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 632);