Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C53×#405 ⋊ C2 |
| Sylow | C53 ⋊ #3155 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#405 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #3155 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|b^2=eg,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^52,d^c=dg1072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27302360, #27302362, #27302974, … |
| maximal normal subgroup of | #27302360, #27302362, #27302974, … |
| maximal quotient of | #27371957, #27372001, #27372022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 14996);