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 ⋊ #1994 |
284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,11)(5,8,9,10);(2,3)(13,17,19,18);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,18,16,19,14,17)Polycyclic presentation:
2^7,3|a^2=f,b^2=deg,c^2=ef,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=cef,d^a=dg,h^a=h^2,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10687244, #10687247, #10687743, … |
| maximal normal subgroup of | #10687244, #10687247, #10687743, … |
| maximal quotient of | #10695195, #10695199, #10695219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4415);