Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #393×C181 ⋊ C2 |
| Sylow | C181 ⋊ #2398 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D181 |
| commutator, socle, cosocle | C23×C181 | C24 |
| Fitting | #393×C181 | C2 |
| prime radical | C22×C22⋊D181 | C22 |
| hypercenter | #393 | D181 |
| hypocenter | C181 | #2398 |
| LCS | G ▷ C23×C181 ▷ C181 | |
| UCS | C1 ◁ C23 ◁ #393 | |
568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 11584 |
Polycyclic presentation:
2^7,181|a^2=e,b^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^d=h^1802024 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 7792);