Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×H(3,ℤ/4) ⋊ C2 |
| Sylow | C3 ⋊ #1339 |
114 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 31 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:
(2,3)(5,9)(6,15)(7,16)(10,12)(11,18)(13,14)(17,19);(1,2,3)(4,5)(6,14,17,9)(7,11)(8,13)(10,15,18,19)(12,16)Polycyclic presentation:
2^7,3|a^2=d,c^2=fg,e^2=g,f^2=g,b^a=bc,c^a=ce,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fg,h^b=h^2,d^c=dg42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1339, C2×#67481, #10705335, … |
| maximal normal subgroup of | D3×#1339, C2×#67481, #10705335, … |
| maximal quotient of | #10681859, #10681862, #10681871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 57);