Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#475 ⋊ C2 |
| Sylow | C3 ⋊ Q8×SD16 |
370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,5)(6,9,11,10);(2,3)(13,16)(15,19)(17,18);(2,3)(4,6,8,11)(5,9,7,10);(4,5,8,7)(6,10,11,9)(12,13,14,17)(15,16,19,18)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,d^2=eg,f^2=g,b^a=beg,c^a=cf,f^a=fg,d^b=deg,h^b=h^2,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10784682, #10784691, #10784742, … |
| maximal normal subgroup of | #10784682, #10784691, #10784742, … |
| maximal quotient of | #10811775, #10811789, #10813583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13615);