Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #137×C53 ⋊ C4 |
| Sylow | C53 ⋊ #2947 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×#6157 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C4 | C23×#6157 |
| Fitting | #137×C53 | C4 |
| socle | C2×C53 | C2×C4×D4 |
| hypercenter | #137 | #6157 |
| hypocenter | C53 | #2947 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #137 | |
470 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=f,b^2=g,c^2=f,e^2=g,b^a=be,e^a=eg,h^a=h^23,e^b=eg,d^c=dg,h^c=h^30,h^f=h^52187 conjugacy classes — view the full character table.
| maximal subgroup of | #28325872, #28325874, #28325876, … |
| maximal normal subgroup of | #28325872, #28325874, #28325876, … |
| maximal quotient of | #28316400, #28316401, #28316402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21268);