Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×Q8⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1623 |
228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 19 points:
(1,2,3)(12,13,15,17)(14,19,18,16);(2,3)(4,5,7,9)(6,11,10,8)(14,18)(16,19);(2,3)(5,8)(6,10)(9,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^7,3|b^2=f,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1623, C2×#68264, #10771084, … |
| maximal normal subgroup of | D3×#1623, C2×#68264, #10771084, … |
| maximal quotient of | #10683929, #10683967, #10683968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 840);