Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C109 ⋊ C8×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#89137 |
| commutator | C2×C109 | C22×C8 |
| Fitting | C22⋊C4×C109 | C4 |
| socle | C22×C109 | C22×C4 |
| hypercenter | C22⋊C4 | #89137 |
| hypocenter | C109 | C8×D4 |
| LCS | G ▷ C2×C109 ▷ C109 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
134 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 56 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=e,c^2=e,e^2=f,b^a=bd,g^a=g^76,g^c=g^33,g^e=g^108310 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20296053, D4×D109⋊C8, #28379746, … |
| maximal normal subgroup of | C2×#20296053, D4×D109⋊C8, #28379746, … |
| maximal quotient of | #28378612, #28378615, #28378616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 1576);