Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C29×C4⋊Q8 ⋊ C4 |
| Sylow | C29 ⋊ #2327 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#1181 |
| commutator, socle | C22×C29 | C23×C4 |
| Fitting | C29×C4⋊Q8 | C4 |
| prime radical, cosocle | C22×D29 | C24 |
| hypercenter | C4⋊Q8 | #1181 |
| hypocenter | C29 | #2327 |
| LCS | G ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
516 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 170 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=g,b^2=g,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^17,c^b=ce,h^b=h^12,h^g=h^28142 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 21001);