Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C3⋊#1304 |
| Sylow | C3 ⋊ #1304 |
396 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 396 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) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,6)(5,8)(7,11)(9,10)(12,14)(13,16,17,18)(15,19);(2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,b^a=bc,c^a=cf,e^a=eg,h^a=h^2,c^b=cg,d^b=df54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682670, #10682671, #10682766, … |
| maximal normal subgroup of | #10682670, #10682671, #10682766, … |
| maximal quotient of | #10681757, #10681758, #10681764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 199);