Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#388 ⋊ C2 |
| Sylow | C53 ⋊ #2854 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#388 | C2 |
| hypercenter | #388 | D53 |
| hypocenter | C53 | #2854 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #388 | |
688 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|b^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dg,h^c=h^52772 conjugacy classes — view the full character table.
| maximal subgroup of | #27556664, #27556670, #27556686, … |
| maximal normal subgroup of | #27556664, #27556670, #27556686, … |
| maximal quotient of | #27315272, #27315320, #27315326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 12837);