Finite Group DatabasebetaFinite Group Databasebeta| order | 7456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×Q8×C233 ⋊ C2 |
| Sylow | C233 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D233 |
| commutator, socle | C22×C233 | C23 |
| Fitting | C2×Q8×C233 | C2 |
| hypercenter | C2×Q8 | D233 |
| hypocenter | C233 | #124 |
| LCS | G ▷ C22×C233 ▷ C233 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
76 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,233|b^2=e,c^2=d,b^a=bd,c^a=ce,f^a=f^232,f^b=f^232,f^c=f^2321174 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #28416231, #28416608, #28416610, … |
| maximal normal subgroup of | #28416231, #28416608, #28416610, … |
| maximal quotient of | #28416300, #28416325, #28416336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7456, 135);