Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C241 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D241 |
| commutator, socle | C22×C241 | C23 |
| Fitting | C2×D4×C241 | C2 |
| hypercenter | C2×D4 | D241 |
| hypocenter | C241 | #124 |
| LCS | G ▷ C22×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
76 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|b^2=e,c^2=d,b^a=bd,c^a=ce,f^b=f^240,f^c=f^2401214 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #29507511, #29507888, #29507895, … |
| maximal normal subgroup of | #29507511, #29507888, #29507895, … |
| maximal quotient of | #29507582, #29507611, #29507620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 133);