Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1278 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊D53 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C2×C42 | C2×D53 |
| Fitting | C53×#431 | C2 |
| socle | C22×C53 | #95 |
| hypercenter | #431 | D53 |
| hypocenter | C53 | #1278 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ #431 | |
172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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=ef,h^b=h^521096 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#1278, C2×#20272551, #27273586, … |
| maximal normal subgroup of | D53×#1278, C2×#20272551, #27273586, … |
| maximal quotient of | #27270619, #27270631, #27270637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2346);