hyperelliptic(.ncag).info

the classification of hyperelliptic varieties in dimensions 2, 3 and 4

hyperelliptic groups

Every finite group occurring as the holonomy group of a hyperelliptic variety in dimension $2$, $3$ or $4$, with its character table and the tangent representations it carries. The last three columns count those representations by dimension.

Groups are identified by their GAP SmallGroup id, which is also the address of the page: $[12,3]$ lives at /group/12-3. The last two columns are that same group elsewhere: on the LMFDB, whose label is the SmallGroup id, and on GroupNames, under its name there. Type an id or either name below to jump to a group.

groupSmallGroup id$\#G$classesdim $2$dim $3$dim $4$LMFDBGroupNames
$\mathrm{C}_{2}$$[2, 1]$221232.1C2
$\mathrm{C}_{3}$$[3, 1]$331353.1C3
$\mathrm{C}_{4}$$[4, 1]$441494.1C4
$\mathrm{C}_{2} \times \mathrm{C}_{2}$$[4, 2]$44·244.2C2^2
$\mathrm{C}_{5}$$[5, 1]$55·115.1C5
$\mathrm{S}_3$$[6, 1]$63··36.1S3
$\mathrm{C}_{6}$$[6, 2]$6617236.2C6
$\mathrm{C}_{7}$$[7, 1]$77··27.1C7
$\mathrm{Q}_8$$[8, 4]$85··28.4Q8
$\mathrm{D}_4$$[8, 3]$85·158.3D4
$\mathrm{C}_{8}$$[8, 1]$88·268.1C8
$\mathrm{C}_{2} \times \mathrm{C}_{2} \times \mathrm{C}_{2}$$[8, 5]$88··28.5C2^3
$\mathrm{C}_{4} \times \mathrm{C}_{2}$$[8, 2]$88·2138.2C2xC4
$\mathrm{C}_{3} \times \mathrm{C}_{3}$$[9, 2]$99·169.2C3^2
$\mathrm{C}_{9}$$[9, 1]$99··29.1C9
$\mathrm{C}_{10}$$[10, 2]$1010·1310.2C10
$\mathrm{C}_{12}$$[12, 2]$1212·42712.2C12
$\mathrm{A}_4$$[12, 3]$124··212.3A4
$\mathrm{S}_3 \times \mathrm{C}_2$$[12, 4]$126··712.4D6
$\mathrm{G}(3,4,2)$$[12, 1]$126··512.1Dic3
$\mathrm{C}_{6} \times \mathrm{C}_{2}$$[12, 5]$1212·32512.5C2xC6
$\mathrm{C}_{14}$$[14, 2]$1414··214.2C14
$\mathrm{C}_{15}$$[15, 1]$1515··115.1C15
$\mathrm{C}_{4} \times \mathrm{C}_{2} \times \mathrm{C}_{2}$$[16, 10]$1616··416.10C2^2xC4
$\mathrm{C}_{2} \times \mathrm{D}_4$$[16, 11]$1610··216.11C2xD4
$\mathrm{C}_{4} \times \mathrm{C}_{4}$$[16, 2]$1616·1916.2C4^2
$\mathrm{G}(8,2,5)$$[16, 6]$1610··516.6M4(2)
$\mathrm{D}_4 \circ \mathrm{C}_4$$[16, 13]$1610··516.13C4oD4
$\mathrm{C}_{8} \times \mathrm{C}_{2}$$[16, 5]$1616··416.5C2xC8
$\mathrm{G}(8,2,3)$$[16, 8]$167··516.8SD16
$(\mathrm{C}_4 \times \mathrm{C}_2) \rtimes \mathrm{C}_2$$[16, 3]$1610··216.3C2^2:C4
$\mathrm{G}(4,4,3)$$[16, 4]$1610··216.4C4:C4
$\mathrm{S}_3 \times \mathrm{C}_3$$[18, 3]$189··1018.3C3xS3
$\mathrm{C}_{18}$$[18, 2]$1818··218.2C18
$\mathrm{C}_{6} \times \mathrm{C}_{3}$$[18, 5]$1818·22318.5C3xC6
$\mathrm{C}_{20}$$[20, 2]$2020··220.2C20
$\mathrm{C}_{10} \times \mathrm{C}_{2}$$[20, 5]$2020··120.5C2xC10
$\mathrm{G}(3,8,2)$$[24, 1]$2412··324.1C3:C8
$\mathrm{S}_3 \times \mathrm{C}_4$$[24, 5]$2412··1124.5C4xS3
$(\mathrm{C}_2 \times \mathrm{C}_6) \rtimes \mathrm{C}_2$$[24, 8]$249··524.8C3:D4
$\mathrm{C}_{12} \times \mathrm{C}_{2}$$[24, 9]$2424·12624.9C2xC12
$\mathrm{C}_{2} \times \mathrm{A}_4$$[24, 13]$248··324.13C2xA4
$\mathrm{C}_{3} \times \mathrm{D}_4$$[24, 10]$2415··1524.10C3xD4
$\mathrm{C}_{6} \times \mathrm{C}_{2} \times \mathrm{C}_{2}$$[24, 15]$2424··624.15C2^2xC6
$\mathrm{C}_{3} \times \mathrm{Q}_8$$[24, 11]$2415··824.11C3xQ8
$\mathrm{C}_{24}$$[24, 2]$2424··424.2C24
$\mathrm{C}_{3} \times \mathrm{C}_{3} \times \mathrm{C}_{3}$$[27, 5]$2727··127.5C3^3
$\mathrm{Heis}(3)$$[27, 3]$2711··227.3He3
$\mathrm{C}_{30}$$[30, 4]$3030··330.4C30
$\mathrm{G}(8,4,5) \times \mathrm{C}_2$$[32, 37]$3220··232.37C2xM4(2)
$(\mathrm{C}_4 \times \mathrm{C}_4) \rtimes \mathrm{C}_2$$[32, 24]$3220··132.24C4^2:C2
$\mathrm{C}_{4} \times \mathrm{C}_{4} \times \mathrm{C}_{2}$$[32, 21]$3232··332.21C2xC4^2
$(\mathrm{C}_4 \times \mathrm{C}_4) \rtimes \mathrm{C}_2$$[32, 11]$3214··832.11C4wrC2
$\mathrm{C}_{8} \times \mathrm{C}_{4}$$[32, 3]$3232··232.3C4xC8
$\mathrm{G}(8,4,5)$$[32, 4]$3220··132.4C8:C4
$\mathrm{G}(3,4,2) \times \mathrm{C}_3$$[36, 6]$3618··1036.6C3xDic3
$\mathrm{C}_{6} \times \mathrm{C}_{6}$$[36, 14]$3636·12336.14C6^2
$\mathrm{S}_3 \times \mathrm{C}_6$$[36, 12]$3618··2036.12S3xC6
$\mathrm{C}_{12} \times \mathrm{C}_{3}$$[36, 8]$3636··736.8C3xC12
$\mathrm{C}_{20} \times \mathrm{C}_{2}$$[40, 9]$4040··140.9C2xC20
$((\mathrm{C}_4 \times \mathrm{C}_2) \rtimes \mathrm{C}_2) \times \mathrm{C}_3$$[48, 21]$4830··148.21C3xC2^2:C4
$\mathrm{C}_{12} \times \mathrm{C}_{2} \times \mathrm{C}_{2}$$[48, 44]$4848··448.44C2^2xC12
$\mathrm{G}(4,4,3) \times \mathrm{C}_3$$[48, 22]$4830··148.22C3xC4:C4
$\mathrm{C}_{24} \times \mathrm{C}_{2}$$[48, 23]$4848··248.23C2xC24
$\mathrm{C}_{4} \times \mathrm{A}_4$$[48, 31]$4816··248.31C4xA4
$\mathrm{C}_{12} \times \mathrm{C}_{4}$$[48, 20]$4848··548.20C4xC12
$\mathrm{S}_3 \times \mathrm{C}_3 \times \mathrm{C}_3$$[54, 12]$5427··254.12S3xC3^2
$\mathrm{C}_{6} \times \mathrm{C}_{3} \times \mathrm{C}_{3}$$[54, 15]$5454··354.15C3^2xC6
$\mathrm{C}_{30} \times \mathrm{C}_{2}$$[60, 13]$6060··160.13C2xC30
$\mathrm{C}_{6} \times \mathrm{C}_{6} \times \mathrm{C}_{2}$$[72, 50]$7272··572.50C2xC6^2
$\mathrm{G}(3,8,2) \times \mathrm{C}_3$$[72, 12]$7236··272.12C3xC3:C8
$((\mathrm{C}_2 \times \mathrm{C}_6) \rtimes \mathrm{C}_2) \times \mathrm{C}_3$$[72, 30]$7227··1672.30C3xC3:D4
$\mathrm{S}_3 \times \mathrm{C}_{12}$$[72, 27]$7236··672.27S3xC12
$\mathrm{C}_{12} \times \mathrm{C}_{6}$$[72, 36]$7272··672.36C6xC12
$\mathrm{C}_{12} \times \mathrm{C}_{4} \times \mathrm{C}_{2}$$[96, 161]$9696··196.161C2xC4xC12
$\mathrm{S}_3 \times \mathrm{C}_6 \times \mathrm{C}_3$$[108, 42]$10854··4108.42S3xC3xC6
$\mathrm{C}_{6} \times \mathrm{C}_{6} \times \mathrm{C}_{3}$$[108, 45]$108108··3108.45C3xC6^2
$\mathrm{G}(3,4,2) \times \mathrm{C}_3 \times \mathrm{C}_3$$[108, 32]$10854··2108.32C3^2xDic3
$\mathrm{C}_{12} \times \mathrm{C}_{6} \times \mathrm{C}_{2}$$[144, 178]$144144··1144.178C2xC6xC12