Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1409 |
110 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 23 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)(4,6,8,15)(5,9,12,19)(7,16,17,14)(10,18,11,13);(1,2,3)(4,5)(6,13,14,9)(7,11,17,10)(8,12)(15,18,16,19)Polycyclic presentation:
2^7,3|a^2=d,b^2=g,c^2=fg,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cfg,d^b=def,e^b=eg,h^b=h^2,d^c=dfg,e^d=eg30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1409, C2×#67540, #10715496, … |
| maximal normal subgroup of | D3×#1409, C2×#67540, #10715496, … |
| maximal quotient of | #10682571, #10682603, #10682696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 116);