Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#405 ⋊ C2 |
| Sylow | C53 ⋊ #2955 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D212 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#405 | C2 |
| socle | C2×C53 | C2×C4×D4 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #2955 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ #405 | |
456 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 185 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=f,c^2=f,e^2=g,f^2=g,b^a=be,d^a=dg,e^a=eg,h^a=h^52,e^b=eg,h^b=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27300961, #27300963, #27301007, … |
| maximal normal subgroup of | #27300961, #27300963, #27301007, … |
| maximal quotient of | C8×#13133245, C8×#13133250, #27353679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13414);