Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#349 ⋊ C2 |
| Sylow | C3 ⋊ #1341 |
110 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 110 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:
(1,2)(5,10)(6,14)(7,8)(11,12)(13,19)(15,16)(17,18);(2,3)(4,5,7,12,9,13,8,17)(6,10,14,11,16,19,15,18)Polycyclic presentation:
2^7,3|a^2=d,c^2=fg,d^2=g,e^2=g,f^2=g,b^a=bc,c^a=ce,f^a=fg,h^a=h^2,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×#1341, C2×#67535, #10705349, … |
| maximal normal subgroup of | D3×#1341, C2×#67535, #10705349, … |
| maximal quotient of | #10681885, #10681888, #10681897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 111);