Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C241 ⋊ #104 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊D241 |
| commutator | C4×C241 | C2×C4 |
| Frattini | C2×C4 | C2×D241 |
| Fitting | #41×C241 | C2 |
| socle | C2×C241 | C22⋊C4 |
| hypercenter | #41 | D241 |
| hypocenter | C241 | #104 |
| LCS | G ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
44 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=d,c^2=e,d^2=e,b^a=bc,c^a=ce,f^a=f^240,c^b=ce1214 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #104×D241, C2×#21440569, #29507732, … |
| maximal normal subgroup of | #104×D241, C2×#21440569, #29507732, … |
| maximal quotient of | D4⋊#14515349, Q8⋊#14515349, #29507451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 73);