BUG: GroupBy float16 indexes are not supported

[BohdanBilonoh]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd
import numpy as np

df = pd.DataFrame({
    "column1": np.array([0.1, 0.1, 0.1, 0.2, 0.2, 0.2], dtype=np.float16),
    "column2": [1,2,3,4,5,6]
})

df.groupby("column1").groups

Issue Description

GroupBy fails with float16 key type

Expected Behavior

GroupBy should support float16 key dtype

Installed Versions

INSTALLED VERSIONS

commit : fc37c83
python : 3.11.8.final.0
python-bits : 64
OS : Linux
OS-release : 5.15.0-94-generic
Version : #104~20.04.1-Ubuntu SMP Tue Jan 16 13:34:09 UTC 2024
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : en_US.UTF-8
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 3.0.0.dev0+407.gfc37c83c2d
numpy : 1.26.4
pytz : 2024.1
dateutil : 2.8.2
setuptools : 69.1.0
pip : 24.0
Cython : 3.0.0
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : None
IPython : 8.21.0
pandas_datareader : None
adbc-driver-postgresql: None
adbc-driver-sqlite : None
bs4 : None
bottleneck : None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : None
pyarrow : 15.0.0
pyreadstat : None
python-calamine : None
pyxlsb : None
s3fs : None
scipy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : None
zstandard : None
tzdata : 2024.1
qtpy : None
pyqt5 : None

[phofl]

float16 is not supported In pandas, you'll have a bad time using it

[BohdanBilonoh]

Thank you for clarification!