Finite Group DatabasebetaFinite Group Databasebeta| order | 7164 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊Dic597 |
| unambiguous semidirect product | C199⋊(C3⋊Dic3) |
| derived, Hall | C32×C199 ⋊ C4 |
| Sylow | C199 ⋊ C3⋊Dic3 |
| Sylow | C32 ⋊ Dic199 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D597 |
| commutator, hypocenter | C32×C199 | C4 |
| Fitting, socle | C2×C32×C199 | C2 |
| LCS | G ▷ C32×C199 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,199,2|a^2=e,b^a=b^2,c^a=c^2,d^a=d^1981794 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3⋊Dic2985, C5×C3⋊Dic597, C3⋊(C3⋊(C199⋊AGL(1,𝔽5))) |
| maximal normal subgroup of | C5×C3⋊Dic597 |
| maximal quotient of | C3⋊Dic2985, C5×C3⋊Dic597, C3⋊(C3⋊(C199⋊AGL(1,𝔽5))) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7164, 27);