Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #135×C109 ⋊ C2 |
| Sylow | C109 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D109 |
| commutator | C4×C109 | C24 |
| Frattini | C4 | C23×D109 |
| Fitting | #135×C109 | C2 |
| socle | C2×C109 | C22×D4 |
| hypercenter | #135 | D109 |
| hypocenter | C109 | #577 |
| LCS | G ▷ C4×C109 ▷ C2×C109 ▷ C109 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,109|a^2=f,e^2=f,b^a=be,e^a=ef,e^b=ef,d^c=df,g^c=g^108,g^d=g^108778 conjugacy classes — view the full character table.
| maximal subgroup of | #28374796, #28374799, #28374803, … |
| maximal normal subgroup of | #28374796, #28374799, #28374803, … |
| maximal quotient of | #28371554, #28371561, #28371624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 1356);