Finite Group DatabasebetaFinite Group Databasebeta| order | 2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊Dic137 |
| Sylow | C137 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D137 |
| commutator | C2×C137 | C2×C4 |
| Fitting, prime radical | C23×C137 | C2 |
| socle, cosocle | C22×C137 | C22 |
| hypercenter | C23 | D137 |
| hypocenter | C137 | C22⋊C4 |
| LCS | G ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ C23 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1096 |
Polycyclic presentation:
2^4,137|a^2=d,b^a=bc,e^a=e^136554 conjugacy classes — view the full character table.
| maximal subgroup of | #17984002, #17984003, C22⋊C4×D137, … |
| maximal normal subgroup of | #17984002, #17984003, C22⋊C4×D137, … |
| maximal quotient of | #17983995, C22⋊#11764040, #17984003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2192, 18);