Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1506 |
228 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 73 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)(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(5,8,9,10)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,23,25)(24,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=f,d^2=g,b^a=bd,d^a=df,c^b=cf,d^b=dg,e^b=efg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10687045, #10687046, #10687059, … |
| maximal normal subgroup of | #10687045, #10687046, #10687059, … |
| maximal quotient of | #10683212, #10683246, #10683255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2009);