Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C29 ⋊ #3407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×#1181 |
| commutator, socle | C2×C29 | C23×C8 |
| Fitting | C29×#131 | C4 |
| prime radical | #41×D29 | C4 |
| cosocle | #6597 | C24 |
| hypercenter | #131 | #1181 |
| hypocenter | C29 | #3407 |
| LCS | G ▷ C2×C29 ▷ C29 | |
| UCS | C1 ◁ C4 ◁ #131 | |
360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 928 |
Polycyclic presentation:
2^7,29|a^2=e,e^2=f,f^2=g,d^a=dg,h^a=h^12,c^b=cg,h^d=h^28,h^e=h^28208 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 21478);