Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ C8×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D53 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | C53×#431 | C2 |
| socle | C22×C53 | C2×C42 |
| hypercenter | #431 | D53 |
| hypocenter | C53 | C8×M16 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ #431 | |
224 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 137 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,d^2=f,e^2=g,c^a=cf,h^a=h^52,h^b=h^521120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27274649, #27274653, #27274654, … |
| maximal normal subgroup of | #27274649, #27274653, #27274654, … |
| maximal quotient of | #27273446, #27273475, #27274519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2791);