Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×#89137)⋊C8 |
| Fitting | C2×C8×C109 ⋊ C4 |
| Sylow | C109 ⋊ #336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×#89137 |
| commutator | C2×C109 | C4×C8 |
| Fitting | C2×C8×C109 | C4 |
| socle | C22×C109 | C42 |
| hypercenter | C2×C8 | #89137 |
| hypocenter | C109 | #336 |
| LCS | G ▷ C2×C109 ▷ C109 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
118 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,109|a^2=d,b^2=e,d^2=f,b^a=bc,g^b=g^33,g^e=g^108472 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(C2×#89137)⋊C8, #28378694, #28378697, … |
| maximal normal subgroup of | C2×(C2×#89137)⋊C8, #28378694, #28378697, … |
| maximal quotient of | #28377854, #28377855, #28377857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 1414);