Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #1302 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | (C2×#6157)⋊C4 |
| commutator | C2×C4×C53 | C42 |
| Frattini | C42 | C2×#6157 |
| Fitting | #97×C53 | C4 |
| socle | C22×C53 | #95 |
| hypercenter | #97 | #6157 |
| hypocenter | C53 | #1302 |
| LCS | G ▷ C2×C4×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #97 | |
144 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=d,b^2=e,c^2=f,d^2=g,e^2=g,b^a=bc,c^a=cf,e^a=efg,h^a=h^30,c^b=cg,h^d=h^52298 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20289609, #28301216, #28301221, … |
| maximal normal subgroup of | C2×#20289609, #28301216, #28301221, … |
| maximal quotient of | #28298028, #28298035, #28298037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19404);