direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C29, Q8⋊C58, C8⋊2C58, D4.C58, C232⋊6C2, C58.15D4, C116.18C22, C4.2(C2×C58), (Q8×C29)⋊4C2, C2.4(D4×C29), (D4×C29).2C2, SmallGroup(464,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C29
G = < a,b,c | a29=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of SD16×C29
►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)
(1 153 217 98 52 183 129 60)(2 154 218 99 53 184 130 61)(3 155 219 100 54 185 131 62)(4 156 220 101 55 186 132 63)(5 157 221 102 56 187 133 64)(6 158 222 103 57 188 134 65)(7 159 223 104 58 189 135 66)(8 160 224 105 30 190 136 67)(9 161 225 106 31 191 137 68)(10 162 226 107 32 192 138 69)(11 163 227 108 33 193 139 70)(12 164 228 109 34 194 140 71)(13 165 229 110 35 195 141 72)(14 166 230 111 36 196 142 73)(15 167 231 112 37 197 143 74)(16 168 232 113 38 198 144 75)(17 169 204 114 39 199 145 76)(18 170 205 115 40 200 117 77)(19 171 206 116 41 201 118 78)(20 172 207 88 42 202 119 79)(21 173 208 89 43 203 120 80)(22 174 209 90 44 175 121 81)(23 146 210 91 45 176 122 82)(24 147 211 92 46 177 123 83)(25 148 212 93 47 178 124 84)(26 149 213 94 48 179 125 85)(27 150 214 95 49 180 126 86)(28 151 215 96 50 181 127 87)(29 152 216 97 51 182 128 59)
(59 182)(60 183)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 194)(72 195)(73 196)(74 197)(75 198)(76 199)(77 200)(78 201)(79 202)(80 203)(81 175)(82 176)(83 177)(84 178)(85 179)(86 180)(87 181)(88 172)(89 173)(90 174)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)(101 156)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(109 164)(110 165)(111 166)(112 167)(113 168)(114 169)(115 170)(116 171)(117 205)(118 206)(119 207)(120 208)(121 209)(122 210)(123 211)(124 212)(125 213)(126 214)(127 215)(128 216)(129 217)(130 218)(131 219)(132 220)(133 221)(134 222)(135 223)(136 224)(137 225)(138 226)(139 227)(140 228)(141 229)(142 230)(143 231)(144 232)(145 204)
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), (1,153,217,98,52,183,129,60)(2,154,218,99,53,184,130,61)(3,155,219,100,54,185,131,62)(4,156,220,101,55,186,132,63)(5,157,221,102,56,187,133,64)(6,158,222,103,57,188,134,65)(7,159,223,104,58,189,135,66)(8,160,224,105,30,190,136,67)(9,161,225,106,31,191,137,68)(10,162,226,107,32,192,138,69)(11,163,227,108,33,193,139,70)(12,164,228,109,34,194,140,71)(13,165,229,110,35,195,141,72)(14,166,230,111,36,196,142,73)(15,167,231,112,37,197,143,74)(16,168,232,113,38,198,144,75)(17,169,204,114,39,199,145,76)(18,170,205,115,40,200,117,77)(19,171,206,116,41,201,118,78)(20,172,207,88,42,202,119,79)(21,173,208,89,43,203,120,80)(22,174,209,90,44,175,121,81)(23,146,210,91,45,176,122,82)(24,147,211,92,46,177,123,83)(25,148,212,93,47,178,124,84)(26,149,213,94,48,179,125,85)(27,150,214,95,49,180,126,86)(28,151,215,96,50,181,127,87)(29,152,216,97,51,182,128,59), (59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,172)(89,173)(90,174)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,205)(118,206)(119,207)(120,208)(121,209)(122,210)(123,211)(124,212)(125,213)(126,214)(127,215)(128,216)(129,217)(130,218)(131,219)(132,220)(133,221)(134,222)(135,223)(136,224)(137,225)(138,226)(139,227)(140,228)(141,229)(142,230)(143,231)(144,232)(145,204)>;
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), (1,153,217,98,52,183,129,60)(2,154,218,99,53,184,130,61)(3,155,219,100,54,185,131,62)(4,156,220,101,55,186,132,63)(5,157,221,102,56,187,133,64)(6,158,222,103,57,188,134,65)(7,159,223,104,58,189,135,66)(8,160,224,105,30,190,136,67)(9,161,225,106,31,191,137,68)(10,162,226,107,32,192,138,69)(11,163,227,108,33,193,139,70)(12,164,228,109,34,194,140,71)(13,165,229,110,35,195,141,72)(14,166,230,111,36,196,142,73)(15,167,231,112,37,197,143,74)(16,168,232,113,38,198,144,75)(17,169,204,114,39,199,145,76)(18,170,205,115,40,200,117,77)(19,171,206,116,41,201,118,78)(20,172,207,88,42,202,119,79)(21,173,208,89,43,203,120,80)(22,174,209,90,44,175,121,81)(23,146,210,91,45,176,122,82)(24,147,211,92,46,177,123,83)(25,148,212,93,47,178,124,84)(26,149,213,94,48,179,125,85)(27,150,214,95,49,180,126,86)(28,151,215,96,50,181,127,87)(29,152,216,97,51,182,128,59), (59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,172)(89,173)(90,174)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)(101,156)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,205)(118,206)(119,207)(120,208)(121,209)(122,210)(123,211)(124,212)(125,213)(126,214)(127,215)(128,216)(129,217)(130,218)(131,219)(132,220)(133,221)(134,222)(135,223)(136,224)(137,225)(138,226)(139,227)(140,228)(141,229)(142,230)(143,231)(144,232)(145,204) );
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)], [(1,153,217,98,52,183,129,60),(2,154,218,99,53,184,130,61),(3,155,219,100,54,185,131,62),(4,156,220,101,55,186,132,63),(5,157,221,102,56,187,133,64),(6,158,222,103,57,188,134,65),(7,159,223,104,58,189,135,66),(8,160,224,105,30,190,136,67),(9,161,225,106,31,191,137,68),(10,162,226,107,32,192,138,69),(11,163,227,108,33,193,139,70),(12,164,228,109,34,194,140,71),(13,165,229,110,35,195,141,72),(14,166,230,111,36,196,142,73),(15,167,231,112,37,197,143,74),(16,168,232,113,38,198,144,75),(17,169,204,114,39,199,145,76),(18,170,205,115,40,200,117,77),(19,171,206,116,41,201,118,78),(20,172,207,88,42,202,119,79),(21,173,208,89,43,203,120,80),(22,174,209,90,44,175,121,81),(23,146,210,91,45,176,122,82),(24,147,211,92,46,177,123,83),(25,148,212,93,47,178,124,84),(26,149,213,94,48,179,125,85),(27,150,214,95,49,180,126,86),(28,151,215,96,50,181,127,87),(29,152,216,97,51,182,128,59)], [(59,182),(60,183),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,194),(72,195),(73,196),(74,197),(75,198),(76,199),(77,200),(78,201),(79,202),(80,203),(81,175),(82,176),(83,177),(84,178),(85,179),(86,180),(87,181),(88,172),(89,173),(90,174),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155),(101,156),(102,157),(103,158),(104,159),(105,160),(106,161),(107,162),(108,163),(109,164),(110,165),(111,166),(112,167),(113,168),(114,169),(115,170),(116,171),(117,205),(118,206),(119,207),(120,208),(121,209),(122,210),(123,211),(124,212),(125,213),(126,214),(127,215),(128,216),(129,217),(130,218),(131,219),(132,220),(133,221),(134,222),(135,223),(136,224),(137,225),(138,226),(139,227),(140,228),(141,229),(142,230),(143,231),(144,232),(145,204)]])
203 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 29A | ··· | 29AB | 58A | ··· | 58AB | 58AC | ··· | 58BD | 116A | ··· | 116AB | 116AC | ··· | 116BD | 232A | ··· | 232BD |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
203 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C29 | C58 | C58 | C58 | D4 | SD16 | D4×C29 | SD16×C29 |
| kernel | SD16×C29 | C232 | D4×C29 | Q8×C29 | SD16 | C8 | D4 | Q8 | C58 | C29 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 28 | 28 | 28 | 28 | 1 | 2 | 28 | 56 |
Matrix representation of SD16×C29 ►in GL2(𝔽233) generated by
| 128 | 0 |
| 0 | 128 |
| 62 | 171 |
| 62 | 62 |
| 1 | 0 |
| 0 | 232 |
G:=sub<GL(2,GF(233))| [128,0,0,128],[62,62,171,62],[1,0,0,232] >;
SD16×C29 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{29} % in TeX
G:=Group("SD16xC29"); // GroupNames label
G:=SmallGroup(464,26);
// by ID
G=gap.SmallGroup(464,26);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,1160,1181,6963,3488,58]);
// Polycyclic
G:=Group<a,b,c|a^29=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations
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