Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #2315 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C22×C22⋊C4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C22×C22⋊C4 | D53 |
| hypocenter | C53 | #2315 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
816 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 269 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=g,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce,h^c=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27290299, #27290301, #27290303, … |
| maximal normal subgroup of | #27290299, #27290301, #27290303, … |
| maximal quotient of | #27305346, #27305371, #27305375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7122);