Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | #585×C181 ⋊ C2 |
| Sylow | C181 ⋊ H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D181 |
| commutator, socle, cosocle | C2×C181 | C26 |
| Fitting | #585×C181 | C2 |
| hypercenter | #585 | D181 |
| hypocenter | C181 | H(7,𝔽2) |
| LCS | G ▷ C2×C181 ▷ C181 | |
| UCS | C1 ◁ C2 ◁ #585 | |
5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 2896 |
Polycyclic presentation:
2^7,181|b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^e=h^180,h^f=h^1803125 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 19321);