Finite Group DatabasebetaFinite Group Databasebeta| order | 7712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C241 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D241 |
| commutator | C2×C241 | C22×C4 |
| Fitting | C4⋊C4×C241 | C2 |
| socle | C22×C241 | C23 |
| hypercenter | C4⋊C4 | D241 |
| hypocenter | C241 | C4×Q8 |
| LCS | G ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
70 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,241|a^2=e,b^2=d,c^2=de,b^a=bd,f^c=f^2401220 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #29507804, #29507808, #29507809, … |
| maximal normal subgroup of | #29507804, #29507808, #29507809, … |
| maximal quotient of | #29507568, C4⋊C4×Dic241, #29507598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7712, 111);