Finite Group DatabasebetaFinite Group Databasebeta| order | 8144 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊#14523266 |
| unambiguous semidirect product | (C2×#14523266)⋊C2 |
| Fitting | C22×C509 ⋊ C4 |
| Sylow | C509 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#14523266 |
| commutator, socle | C2×C509 | C2×C4 |
| Fitting | C22×C509 | C4 |
| hypercenter | C22 | #14523266 |
| hypocenter | C509 | C22⋊C4 |
| LCS | G ▷ C2×C509 ▷ C509 | |
| UCS | C1 ◁ C2 ◁ C22 | |
34 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,509|a^2=d,b^a=bc,e^a=e^301,e^d=e^508518 conjugacy classes — view the full character table.
| maximal subgroup of | #30654273, #30654274, C2×C22⋊#14523266, … |
| maximal normal subgroup of | #30654273, #30654274, C2×C22⋊#14523266, … |
| maximal quotient of | (C2×#14523266)⋊C4, C22⋊#16861332, #30654272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8144, 45);