Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #1286 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133124 |
| commutator | C2×C4×C53 | C42 |
| Frattini | C2×C42 | C2×D53 |
| Fitting | C53×#423 | C2 |
| socle | C22×C53 | #95 |
| hypercenter | #423 | D53 |
| hypocenter | C53 | #1286 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #423 | |
156 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 73 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=fg,e^2=f,b^a=bc,c^a=cf,e^a=efg,h^a=h^52,d^b=dg,h^b=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1286, C2×#20272575, #27273870, … |
| maximal normal subgroup of | D53×#1286, C2×#20272575, #27273870, … |
| maximal quotient of | #27270625, #27270636, #27270640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2370);