Finite Group DatabasebetaFinite Group Databasebeta| order | 5888 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C23 ⋊ #7526 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #15671549 |
| commutator | C23×#125 | C2×C4 |
| Frattini | #491 | C2×D23 |
| Fitting | C23×#1356 | C2 |
| socle | C2×C23 | #1404 |
| hypercenter | #1356 | D23 |
| hypocenter | C23 | #7526 |
| LCS | G ▷ C23×#125 ▷ C42×C23 ▷ C2×C4×C23 ▷ C22×C23 ▷ C2×C23 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1356 | |
126 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^8,23|a^2=d,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^22,c^b=cg,d^b=deh,e^b=egh,f^b=fg,i^b=i^22,d^c=dfg,e^c=eg,f^c=fh,e^d=egh,f^d=fh239 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5888, 57418);