direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C29, C4.C58, C116.3C2, C58.7C22, C2.2(C2×C58), SmallGroup(232,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C29
G = < a,b,c | a29=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C29
►Regular action on 232 points
Generators in S232
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29)(30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58)(59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87)(88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116)(117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145)(146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174)(175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203)(204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232)
(1 223 114 166)(2 224 115 167)(3 225 116 168)(4 226 88 169)(5 227 89 170)(6 228 90 171)(7 229 91 172)(8 230 92 173)(9 231 93 174)(10 232 94 146)(11 204 95 147)(12 205 96 148)(13 206 97 149)(14 207 98 150)(15 208 99 151)(16 209 100 152)(17 210 101 153)(18 211 102 154)(19 212 103 155)(20 213 104 156)(21 214 105 157)(22 215 106 158)(23 216 107 159)(24 217 108 160)(25 218 109 161)(26 219 110 162)(27 220 111 163)(28 221 112 164)(29 222 113 165)(30 85 178 117)(31 86 179 118)(32 87 180 119)(33 59 181 120)(34 60 182 121)(35 61 183 122)(36 62 184 123)(37 63 185 124)(38 64 186 125)(39 65 187 126)(40 66 188 127)(41 67 189 128)(42 68 190 129)(43 69 191 130)(44 70 192 131)(45 71 193 132)(46 72 194 133)(47 73 195 134)(48 74 196 135)(49 75 197 136)(50 76 198 137)(51 77 199 138)(52 78 200 139)(53 79 201 140)(54 80 202 141)(55 81 203 142)(56 82 175 143)(57 83 176 144)(58 84 177 145)
(1 55 114 203)(2 56 115 175)(3 57 116 176)(4 58 88 177)(5 30 89 178)(6 31 90 179)(7 32 91 180)(8 33 92 181)(9 34 93 182)(10 35 94 183)(11 36 95 184)(12 37 96 185)(13 38 97 186)(14 39 98 187)(15 40 99 188)(16 41 100 189)(17 42 101 190)(18 43 102 191)(19 44 103 192)(20 45 104 193)(21 46 105 194)(22 47 106 195)(23 48 107 196)(24 49 108 197)(25 50 109 198)(26 51 110 199)(27 52 111 200)(28 53 112 201)(29 54 113 202)(59 230 120 173)(60 231 121 174)(61 232 122 146)(62 204 123 147)(63 205 124 148)(64 206 125 149)(65 207 126 150)(66 208 127 151)(67 209 128 152)(68 210 129 153)(69 211 130 154)(70 212 131 155)(71 213 132 156)(72 214 133 157)(73 215 134 158)(74 216 135 159)(75 217 136 160)(76 218 137 161)(77 219 138 162)(78 220 139 163)(79 221 140 164)(80 222 141 165)(81 223 142 166)(82 224 143 167)(83 225 144 168)(84 226 145 169)(85 227 117 170)(86 228 118 171)(87 229 119 172)
G:=sub<Sym(232)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145)(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203)(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,223,114,166)(2,224,115,167)(3,225,116,168)(4,226,88,169)(5,227,89,170)(6,228,90,171)(7,229,91,172)(8,230,92,173)(9,231,93,174)(10,232,94,146)(11,204,95,147)(12,205,96,148)(13,206,97,149)(14,207,98,150)(15,208,99,151)(16,209,100,152)(17,210,101,153)(18,211,102,154)(19,212,103,155)(20,213,104,156)(21,214,105,157)(22,215,106,158)(23,216,107,159)(24,217,108,160)(25,218,109,161)(26,219,110,162)(27,220,111,163)(28,221,112,164)(29,222,113,165)(30,85,178,117)(31,86,179,118)(32,87,180,119)(33,59,181,120)(34,60,182,121)(35,61,183,122)(36,62,184,123)(37,63,185,124)(38,64,186,125)(39,65,187,126)(40,66,188,127)(41,67,189,128)(42,68,190,129)(43,69,191,130)(44,70,192,131)(45,71,193,132)(46,72,194,133)(47,73,195,134)(48,74,196,135)(49,75,197,136)(50,76,198,137)(51,77,199,138)(52,78,200,139)(53,79,201,140)(54,80,202,141)(55,81,203,142)(56,82,175,143)(57,83,176,144)(58,84,177,145), (1,55,114,203)(2,56,115,175)(3,57,116,176)(4,58,88,177)(5,30,89,178)(6,31,90,179)(7,32,91,180)(8,33,92,181)(9,34,93,182)(10,35,94,183)(11,36,95,184)(12,37,96,185)(13,38,97,186)(14,39,98,187)(15,40,99,188)(16,41,100,189)(17,42,101,190)(18,43,102,191)(19,44,103,192)(20,45,104,193)(21,46,105,194)(22,47,106,195)(23,48,107,196)(24,49,108,197)(25,50,109,198)(26,51,110,199)(27,52,111,200)(28,53,112,201)(29,54,113,202)(59,230,120,173)(60,231,121,174)(61,232,122,146)(62,204,123,147)(63,205,124,148)(64,206,125,149)(65,207,126,150)(66,208,127,151)(67,209,128,152)(68,210,129,153)(69,211,130,154)(70,212,131,155)(71,213,132,156)(72,214,133,157)(73,215,134,158)(74,216,135,159)(75,217,136,160)(76,218,137,161)(77,219,138,162)(78,220,139,163)(79,221,140,164)(80,222,141,165)(81,223,142,166)(82,224,143,167)(83,225,144,168)(84,226,145,169)(85,227,117,170)(86,228,118,171)(87,229,119,172)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145)(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203)(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,223,114,166)(2,224,115,167)(3,225,116,168)(4,226,88,169)(5,227,89,170)(6,228,90,171)(7,229,91,172)(8,230,92,173)(9,231,93,174)(10,232,94,146)(11,204,95,147)(12,205,96,148)(13,206,97,149)(14,207,98,150)(15,208,99,151)(16,209,100,152)(17,210,101,153)(18,211,102,154)(19,212,103,155)(20,213,104,156)(21,214,105,157)(22,215,106,158)(23,216,107,159)(24,217,108,160)(25,218,109,161)(26,219,110,162)(27,220,111,163)(28,221,112,164)(29,222,113,165)(30,85,178,117)(31,86,179,118)(32,87,180,119)(33,59,181,120)(34,60,182,121)(35,61,183,122)(36,62,184,123)(37,63,185,124)(38,64,186,125)(39,65,187,126)(40,66,188,127)(41,67,189,128)(42,68,190,129)(43,69,191,130)(44,70,192,131)(45,71,193,132)(46,72,194,133)(47,73,195,134)(48,74,196,135)(49,75,197,136)(50,76,198,137)(51,77,199,138)(52,78,200,139)(53,79,201,140)(54,80,202,141)(55,81,203,142)(56,82,175,143)(57,83,176,144)(58,84,177,145), (1,55,114,203)(2,56,115,175)(3,57,116,176)(4,58,88,177)(5,30,89,178)(6,31,90,179)(7,32,91,180)(8,33,92,181)(9,34,93,182)(10,35,94,183)(11,36,95,184)(12,37,96,185)(13,38,97,186)(14,39,98,187)(15,40,99,188)(16,41,100,189)(17,42,101,190)(18,43,102,191)(19,44,103,192)(20,45,104,193)(21,46,105,194)(22,47,106,195)(23,48,107,196)(24,49,108,197)(25,50,109,198)(26,51,110,199)(27,52,111,200)(28,53,112,201)(29,54,113,202)(59,230,120,173)(60,231,121,174)(61,232,122,146)(62,204,123,147)(63,205,124,148)(64,206,125,149)(65,207,126,150)(66,208,127,151)(67,209,128,152)(68,210,129,153)(69,211,130,154)(70,212,131,155)(71,213,132,156)(72,214,133,157)(73,215,134,158)(74,216,135,159)(75,217,136,160)(76,218,137,161)(77,219,138,162)(78,220,139,163)(79,221,140,164)(80,222,141,165)(81,223,142,166)(82,224,143,167)(83,225,144,168)(84,226,145,169)(85,227,117,170)(86,228,118,171)(87,229,119,172) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29),(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58),(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87),(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116),(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145),(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174),(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203),(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)], [(1,223,114,166),(2,224,115,167),(3,225,116,168),(4,226,88,169),(5,227,89,170),(6,228,90,171),(7,229,91,172),(8,230,92,173),(9,231,93,174),(10,232,94,146),(11,204,95,147),(12,205,96,148),(13,206,97,149),(14,207,98,150),(15,208,99,151),(16,209,100,152),(17,210,101,153),(18,211,102,154),(19,212,103,155),(20,213,104,156),(21,214,105,157),(22,215,106,158),(23,216,107,159),(24,217,108,160),(25,218,109,161),(26,219,110,162),(27,220,111,163),(28,221,112,164),(29,222,113,165),(30,85,178,117),(31,86,179,118),(32,87,180,119),(33,59,181,120),(34,60,182,121),(35,61,183,122),(36,62,184,123),(37,63,185,124),(38,64,186,125),(39,65,187,126),(40,66,188,127),(41,67,189,128),(42,68,190,129),(43,69,191,130),(44,70,192,131),(45,71,193,132),(46,72,194,133),(47,73,195,134),(48,74,196,135),(49,75,197,136),(50,76,198,137),(51,77,199,138),(52,78,200,139),(53,79,201,140),(54,80,202,141),(55,81,203,142),(56,82,175,143),(57,83,176,144),(58,84,177,145)], [(1,55,114,203),(2,56,115,175),(3,57,116,176),(4,58,88,177),(5,30,89,178),(6,31,90,179),(7,32,91,180),(8,33,92,181),(9,34,93,182),(10,35,94,183),(11,36,95,184),(12,37,96,185),(13,38,97,186),(14,39,98,187),(15,40,99,188),(16,41,100,189),(17,42,101,190),(18,43,102,191),(19,44,103,192),(20,45,104,193),(21,46,105,194),(22,47,106,195),(23,48,107,196),(24,49,108,197),(25,50,109,198),(26,51,110,199),(27,52,111,200),(28,53,112,201),(29,54,113,202),(59,230,120,173),(60,231,121,174),(61,232,122,146),(62,204,123,147),(63,205,124,148),(64,206,125,149),(65,207,126,150),(66,208,127,151),(67,209,128,152),(68,210,129,153),(69,211,130,154),(70,212,131,155),(71,213,132,156),(72,214,133,157),(73,215,134,158),(74,216,135,159),(75,217,136,160),(76,218,137,161),(77,219,138,162),(78,220,139,163),(79,221,140,164),(80,222,141,165),(81,223,142,166),(82,224,143,167),(83,225,144,168),(84,226,145,169),(85,227,117,170),(86,228,118,171),(87,229,119,172)]])
Q8×C29 is a maximal subgroup of
Q8⋊D29 C29⋊Q16 Q8⋊2D29
145 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 29A | ··· | 29AB | 58A | ··· | 58AB | 116A | ··· | 116CF |
| order | 1 | 2 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
145 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C29 | C58 | Q8 | Q8×C29 |
| kernel | Q8×C29 | C116 | Q8 | C4 | C29 | C1 |
| # reps | 1 | 3 | 28 | 84 | 1 | 28 |
Matrix representation of Q8×C29 ►in GL2(𝔽233) generated by
| 64 | 0 |
| 0 | 64 |
| 1 | 231 |
| 1 | 232 |
| 13 | 153 |
| 206 | 220 |
G:=sub<GL(2,GF(233))| [64,0,0,64],[1,1,231,232],[13,206,153,220] >;
Q8×C29 in GAP, Magma, Sage, TeX
Q_8\times C_{29} % in TeX
G:=Group("Q8xC29"); // GroupNames label
G:=SmallGroup(232,11);
// by ID
G=gap.SmallGroup(232,11);
# by ID
G:=PCGroup([4,-2,-2,-29,-2,464,945,469]);
// Polycyclic
G:=Group<a,b,c|a^29=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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