Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×Q8×C241 ⋊ C2 |
| Sylow | C241 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C22⋊D241 |
| commutator | C4×C241 | C23 |
| Frattini | C4 | C22×D241 |
| Fitting | C2×Q8×C241 | C2 |
| socle | C2×C241 | C2×D4 |
| hypercenter | C2×Q8 | D241 |
| hypocenter | C241 | #137 |
| LCS | G ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×Q8 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|b^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^240,d^b=de1211 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #29507893, #29507894, #29507898, … |
| maximal normal subgroup of | #29507893, #29507894, #29507898, … |
| maximal quotient of | #29507716, #29507726, #29507757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 177);