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Fibrifold
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<h2 id="irreducible-cubic-space-groups">Irreducible cubic space groups</h2> <p>The 35 irreducible space groups correspond to the <a href="/facts/Cubic_space_group/92Co7BKM">cubic space group</a>. </p> 35 irreducible space groups<table><tbody><tr><td>8o:2</td><td>4−:2</td><td>4o:2</td><td>4+:2</td><td>2−:2</td><td>2o:2</td><td>2+:2</td><td>1o:2</td></tr><tr><td>8o</td><td>4−</td><td>4o</td><td>4+</td><td>2−</td><td>2o</td><td>2+</td><td>1o</td></tr><tr><td>8o/4</td><td>4−/4</td><td>4o/4</td><td>4+/4</td><td>2−/4</td><td>2o/4</td><td>2+/4</td><td>1o/4</td></tr><tr><td>8−o</td><td>8oo</td><td>8+o</td><td>4− −</td><td>4−o</td><td>4oo</td><td>4+o</td><td>4++</td><td>2−o</td><td>2oo</td><td>2+o</td></tr></tbody></table> 36 cubic groups<table><tbody><tr><th>ClassPoint group</th><th>Hexoctahedral*432 (m3m)</th><th>Hextetrahedral*332 (43m)</th><th>Gyroidal432 (432)</th><th>Diploidal3*2 (m3)</th><th>Tetartoidal332 (23)</th></tr><tr><th>bc lattice (I)</th><td>8o:2 (Im3m)</td><td>4o:2 (I43m)</td><td>8+o (I432)</td><td>8−o (I3)</td><td>4oo (I23)</td></tr><tr><th rowspan="2">nc lattice (P)</th><td>4−:2 (Pm3m)</td><td rowspan="2">2o:2 (P43m)</td><td>4−o (P432)</td><td>4− (Pm3)</td><td rowspan="2">2o (P23)</td></tr><tr><td>4+:2 (Pn3m)</td><td>4+ (P4232)</td><td>4+o (Pn3)</td></tr><tr><th rowspan="2">fc lattice (F)</th><td>2−:2 (Fm3m)</td><td rowspan="2">1o:2 (F43m)</td><td>2−o (F432)</td><td>2− (Fm3)</td><td rowspan="2">1o (F23)</td></tr><tr><td>2+:2 (Fd3m)</td><td>2+ (F4132)</td><td>2+o (Fd3)</td></tr><tr><th>Otherlatticegroups</th><td>8o (Pm3n)8oo (Pn3n)4− − (Fm3c)4++ (Fd3c)</td><td>4o (P43n)2oo (F43c)</td><td></td><td></td><td></td></tr><tr><th>Achiralquartergroups</th><td>8o/4 (Ia3d)</td><td>4o/4 (I43d)</td><td>4+/4 (I4132)2+/4 (P4332,P4132)</td><td>2−/4 (Pa3)4−/4 (Ia3)</td><td>1o/4 (P213)2o/4 (I213)</td></tr></tbody></table> <table><tbody><tr><td></td><td></td><td></td></tr><tr><td>8 primary hexoctahedral hextetrahedral lattices of the cubic space groups</td><td colspan="2">The fibrifold cubic subgroup structure shown is based on extending symmetry of the <a href="/facts/Tetragonal_disphenoid/7rU7jc7U">tetragonal disphenoid</a> fundamental domain of space group 216, similar to the <a href="/facts/Square/JHJeMvmL">square</a></td></tr></tbody></table> <p>Irreducible group symbols (indexed 195−230) in <a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin notation</a>, Fibrifold notation, geometric notation, and <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a>: </p> <table><tbody><tr><th>Class(<a href="/facts/Orbifold_notation/HCbBoa2J">Orbifold</a> point group)</th><th colspan="10"><a href="/facts/Space_group/2XQx0J7M">Space groups</a></th></tr><tr><th rowspan="5">Tetartoidal23(332)</th><th>195</th><th>196</th><th>197</th><th>198</th><th>199</th><th colspan="5"> </th></tr><tr><td>P23</td><td>F23</td><td>I23</td><td>P213</td><td>I213</td><td colspan="5"> </td></tr><tr><td>2o</td><td>1o</td><td>4oo</td><td>1o/4</td><td>2o/4</td><td colspan="5"> </td></tr><tr><td>P3.3.2</td><td>F3.3.2</td><td>I3.3.2</td><td>P3.3.21</td><td>I3.3.21</td><td colspan="5"> </td></tr><tr><td>[(4,3+,4,2+)]</td><td>[3[4]]+</td><td>[[(4,3+,4,2+)]]</td><td></td><td></td><td colspan="5"> </td></tr><tr><th rowspan="5">Diploidal43m(3*2)</th><th>200</th><th>201</th><th>202</th><th>203</th><th>204</th><th>205</th><th>206</th><th colspan="3"> </th></tr><tr><td>Pm3</td><td>Pn3</td><td>Fm3</td><td>Fd3</td><td>I3</td><td>Pa3</td><td>Ia3</td><td colspan="3"> </td></tr><tr><td>4−</td><td>4+o</td><td>2−</td><td>2+o</td><td>8−o</td><td>2−/4</td><td>4−/4</td><td colspan="3"> </td></tr><tr><td>P43</td><td>Pn43</td><td>F43</td><td>Fd43</td><td>I43</td><td>Pb43</td><td>Ib43</td><td colspan="3"> </td></tr><tr><td>[4,3+,4]</td><td>[[4,3+,4]+]</td><td>[4,(31,1)+]</td><td>[[3[4]]]+</td><td>[[4,3+,4]]</td><td></td><td></td><td colspan="3"> </td></tr><tr><th rowspan="5">Gyroidal432(432)</th><th>207</th><th>208</th><th>209</th><th>210</th><th>211</th><th>212</th><th>213</th><th>214</th><th colspan="2"> </th></tr><tr><td>P432</td><td>P4232</td><td>F432</td><td>F4132</td><td>I432</td><td>P4332</td><td>P4132</td><td>I4132</td><td colspan="2"> </td></tr><tr><td>4−o</td><td>4+</td><td>2−o</td><td>2+</td><td>8+o</td><td colspan="2">2+/4</td><td>4+/4</td><td colspan="2"> </td></tr><tr><td>P4.3.2</td><td>P42.3.2</td><td>F4.3.2</td><td>F41.3.2</td><td>I4.3.2</td><td>P43.3.2</td><td>P41.3.2</td><td>I41.3.2</td><td colspan="2"> </td></tr><tr><td>[4,3,4]+</td><td>[[4,3,4]+]+</td><td>[4,31,1]+</td><td>[[3[4]]]+</td><td>[[4,3,4]]+</td><td colspan="2"></td><td></td><td colspan="2"> </td></tr><tr><th rowspan="5">Hextetrahedral43m(*332)</th><th>215</th><th>216</th><th>217</th><th>218</th><th>219</th><th>220</th><th colspan="4"> </th></tr><tr><td>P43m</td><td>F43m</td><td>I43m</td><td>P43n</td><td>F43c</td><td>I43d</td><td colspan="4"> </td></tr><tr><td>2o:2</td><td>1o:2</td><td>4o:2</td><td>4o</td><td>2oo</td><td>4o/4</td><td colspan="4"> </td></tr><tr><td>P33</td><td>F33</td><td>I33</td><td>Pn3n3n</td><td>Fc3c3a</td><td>Id3d3d</td><td colspan="4"> </td></tr><tr><td>[(4,3,4,2+)]</td><td>[3[4]]</td><td>[[(4,3,4,2+)]]</td><td>[[(4,3,4,2+)]+]</td><td>[+(4,{3),4}+]</td><td></td><td colspan="4"> </td></tr><tr><th rowspan="5">Hexoctahedralm3m(*432)</th><th>221</th><th>222</th><th>223</th><th>224</th><th>225</th><th>226</th><th>227</th><th>228</th><th>229</th><th>230</th></tr><tr><td>Pm3m</td><td>Pn3n</td><td>Pm3n</td><td>Pn3m</td><td>Fm3m</td><td>Fm3c</td><td>Fd3m</td><td>Fd3c</td><td>Im3m</td><td>Ia3d</td></tr><tr><td>4−:2</td><td>8oo</td><td>8o</td><td>4+:2</td><td>2−:2</td><td>4−−</td><td>2+:2</td><td>4++</td><td>8o:2</td><td>8o/4</td></tr><tr><td>P43</td><td>Pn4n3n</td><td>P4n3n</td><td>Pn43</td><td>F43</td><td>F4c3a</td><td>Fd4n3</td><td>Fd4c3a</td><td>I43</td><td>Ib4d3d</td></tr><tr><td>[4,3,4]</td><td></td><td>[[4,3,4]+]</td><td>[(4+,2+)[3[4]]]</td><td>[4,31,1]</td><td>[4,(3,4)+]</td><td>[[3[4]]]</td><td>[[+(4,{3),4}+]]</td><td>[[4,3,4]]</td><td></td></tr></tbody></table> <ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John Horton</a>; Delgado Friedrichs, Olaf; Huson, Daniel H.; <a href="/facts/William_Thurston/K1Qe8801">Thurston, William P.</a> (2001), <a href="http://www.emis.de/journals/BAG/vol.42/no.2/17.html">"On three-dimensional space groups"</a>, <i>Beiträge zur Algebra und Geometrie</i>, 42 (2): 475–507, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0138-4821">0138-4821</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1865535">1865535</a></li> <li>Hestenes, David; Holt, Jeremy W. (February 2007), <a href="https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf">"The Crystallographic Space Groups in Geometric Algebra"</a> (PDF), <i>Journal of Mathematical Physics</i>, 48 (2): 023514, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2007JMP....48b3514H">2007JMP....48b3514H</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.2426416">10.1063/1.2426416</a></li> <li>Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), <i>The Symmetries of Things</i>, Taylor & Francis, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1173.00001">1173.00001</a></li> <li>Coxeter, H.S.M. (1995), <a href="https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313">"Regular and Semi Regular Polytopes III"</a>, in Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; et al. (eds.), <a href="https://archive.org/details/kaleidoscopessel0000coxe/page/313"><i>Kaleidoscopes: Selected Writings of H.S.M. Coxeter</i></a>, Wiley, pp. <a href="https://archive.org/details/kaleidoscopessel0000coxe/page/313">313–358</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0976.01023">0976.01023</a></li></ul>