Finite Group DatabasebetaFinite Group Databasebeta| order | 7136 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C22⋊D223)⋊C4 |
| Fitting | C22⋊C4×C223 ⋊ C2 |
| Sylow | C223 ⋊ C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D223 |
| commutator | C2×C223 | C22×C4 |
| Fitting | C22⋊C4×C223 | C2 |
| socle | C22×C223 | C23 |
| hypercenter | C22⋊C4 | D223 |
| hypocenter | C223 | C4×D4 |
| LCS | G ▷ C2×C223 ▷ C223 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
94 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,223|a^2=e,c^2=e,b^a=bd,f^c=f^2221130 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×D4×D223, C2×(C22⋊D223)⋊C4, C4×#16813169, … |
| maximal normal subgroup of | C4×D4×D223, C2×(C22⋊D223)⋊C4, C4×#16813169, … |
| maximal quotient of | #28383427, C22⋊C4×Dic223, #28383431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7136, 106);