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 ⋊ C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D241 |
| commutator, socle | C22×C241 | C23 |
| Fitting | C2×Q8×C241 | C2 |
| hypercenter | C2×Q8 | D241 |
| hypocenter | C241 | C22⋊Q8 |
| LCS | G ▷ C22×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
74 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=d,b^2=d,b^a=bd,c^a=ce,f^c=f^2401214 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #29507866, #29507871, #29507876, … |
| maximal normal subgroup of | #29507866, #29507871, #29507876, … |
| maximal quotient of | #29507574, #29507588, #29507598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 122);