Finite Group DatabasebetaFinite Group Databasebeta| order | 1312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | SD16×C41 ⋊ C2 |
| Sylow | C41 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D41 |
| commutator, cosocle | C4×C41 | C23 |
| Frattini | C4 | C22×D41 |
| Fitting | SD16×C41 | C2 |
| socle | C2×C41 | C2×D4 |
| hypercenter | SD16 | D41 |
| hypocenter | C41 | #135 |
| LCS | G ▷ C4×C41 ▷ C2×C41 ▷ C41 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 656 |
Polycyclic presentation:
2^5,41|c^2=e,d^2=e,b^a=bd,d^a=de,d^b=de,f^b=f^40,f^c=f^40154 conjugacy classes — view the full character table.
| maximal subgroup of | #135×D41, C2×#13050185, #14581132, … |
| maximal normal subgroup of | #135×D41, C2×#13050185, #14581132, … |
| maximal quotient of | #14580329, #14580331, #14580371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1312, 167);