Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting | C3×#410 ⋊ C2 |
| Sylow | C3 ⋊ #2128 |
270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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 — 3 generators on 19 points:
(1,2,3)(4,6,8,13)(5,10,12,18)(7,14,15,19)(9,16,11,17);(2,3)(5,9)(6,13)(7,15)(10,17)(11,12)(16,18);(4,5)(6,10)(7,11,15,9)(8,12)(13,18)(14,17,19,16)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=dg,e^b=efg,h^b=h^2,e^d=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10710731, #10710732, #10710746, … |
| maximal normal subgroup of | #10710731, #10710732, #10710746, … |
| maximal quotient of | #10704299, #10704354, #10704475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2637);