direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D29, C232⋊3C2, D58.4C4, C4.12D58, Dic29.4C4, C116.12C22, C29⋊3(C2×C8), C29⋊2C8⋊6C2, C58.8(C2×C4), C2.1(C4×D29), (C4×D29).7C2, SmallGroup(464,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C8×D29 |
Generators and relations for C8×D29
G = < a,b,c | a8=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D29
►On 232 points
Generators in S232
(1 222 106 162 38 202 60 139)(2 223 107 163 39 203 61 140)(3 224 108 164 40 175 62 141)(4 225 109 165 41 176 63 142)(5 226 110 166 42 177 64 143)(6 227 111 167 43 178 65 144)(7 228 112 168 44 179 66 145)(8 229 113 169 45 180 67 117)(9 230 114 170 46 181 68 118)(10 231 115 171 47 182 69 119)(11 232 116 172 48 183 70 120)(12 204 88 173 49 184 71 121)(13 205 89 174 50 185 72 122)(14 206 90 146 51 186 73 123)(15 207 91 147 52 187 74 124)(16 208 92 148 53 188 75 125)(17 209 93 149 54 189 76 126)(18 210 94 150 55 190 77 127)(19 211 95 151 56 191 78 128)(20 212 96 152 57 192 79 129)(21 213 97 153 58 193 80 130)(22 214 98 154 30 194 81 131)(23 215 99 155 31 195 82 132)(24 216 100 156 32 196 83 133)(25 217 101 157 33 197 84 134)(26 218 102 158 34 198 85 135)(27 219 103 159 35 199 86 136)(28 220 104 160 36 200 87 137)(29 221 105 161 37 201 59 138)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(117 154)(118 153)(119 152)(120 151)(121 150)(122 149)(123 148)(124 147)(125 146)(126 174)(127 173)(128 172)(129 171)(130 170)(131 169)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)(181 213)(182 212)(183 211)(184 210)(185 209)(186 208)(187 207)(188 206)(189 205)(190 204)(191 232)(192 231)(193 230)(194 229)(195 228)(196 227)(197 226)(198 225)(199 224)(200 223)(201 222)(202 221)(203 220)
G:=sub<Sym(232)| (1,222,106,162,38,202,60,139)(2,223,107,163,39,203,61,140)(3,224,108,164,40,175,62,141)(4,225,109,165,41,176,63,142)(5,226,110,166,42,177,64,143)(6,227,111,167,43,178,65,144)(7,228,112,168,44,179,66,145)(8,229,113,169,45,180,67,117)(9,230,114,170,46,181,68,118)(10,231,115,171,47,182,69,119)(11,232,116,172,48,183,70,120)(12,204,88,173,49,184,71,121)(13,205,89,174,50,185,72,122)(14,206,90,146,51,186,73,123)(15,207,91,147,52,187,74,124)(16,208,92,148,53,188,75,125)(17,209,93,149,54,189,76,126)(18,210,94,150,55,190,77,127)(19,211,95,151,56,191,78,128)(20,212,96,152,57,192,79,129)(21,213,97,153,58,193,80,130)(22,214,98,154,30,194,81,131)(23,215,99,155,31,195,82,132)(24,216,100,156,32,196,83,133)(25,217,101,157,33,197,84,134)(26,218,102,158,34,198,85,135)(27,219,103,159,35,199,86,136)(28,220,104,160,36,200,87,137)(29,221,105,161,37,201,59,138), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(117,154)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,174)(127,173)(128,172)(129,171)(130,170)(131,169)(132,168)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,211)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,204)(191,232)(192,231)(193,230)(194,229)(195,228)(196,227)(197,226)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)>;
G:=Group( (1,222,106,162,38,202,60,139)(2,223,107,163,39,203,61,140)(3,224,108,164,40,175,62,141)(4,225,109,165,41,176,63,142)(5,226,110,166,42,177,64,143)(6,227,111,167,43,178,65,144)(7,228,112,168,44,179,66,145)(8,229,113,169,45,180,67,117)(9,230,114,170,46,181,68,118)(10,231,115,171,47,182,69,119)(11,232,116,172,48,183,70,120)(12,204,88,173,49,184,71,121)(13,205,89,174,50,185,72,122)(14,206,90,146,51,186,73,123)(15,207,91,147,52,187,74,124)(16,208,92,148,53,188,75,125)(17,209,93,149,54,189,76,126)(18,210,94,150,55,190,77,127)(19,211,95,151,56,191,78,128)(20,212,96,152,57,192,79,129)(21,213,97,153,58,193,80,130)(22,214,98,154,30,194,81,131)(23,215,99,155,31,195,82,132)(24,216,100,156,32,196,83,133)(25,217,101,157,33,197,84,134)(26,218,102,158,34,198,85,135)(27,219,103,159,35,199,86,136)(28,220,104,160,36,200,87,137)(29,221,105,161,37,201,59,138), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(117,154)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,174)(127,173)(128,172)(129,171)(130,170)(131,169)(132,168)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,211)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,204)(191,232)(192,231)(193,230)(194,229)(195,228)(196,227)(197,226)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220) );
G=PermutationGroup([[(1,222,106,162,38,202,60,139),(2,223,107,163,39,203,61,140),(3,224,108,164,40,175,62,141),(4,225,109,165,41,176,63,142),(5,226,110,166,42,177,64,143),(6,227,111,167,43,178,65,144),(7,228,112,168,44,179,66,145),(8,229,113,169,45,180,67,117),(9,230,114,170,46,181,68,118),(10,231,115,171,47,182,69,119),(11,232,116,172,48,183,70,120),(12,204,88,173,49,184,71,121),(13,205,89,174,50,185,72,122),(14,206,90,146,51,186,73,123),(15,207,91,147,52,187,74,124),(16,208,92,148,53,188,75,125),(17,209,93,149,54,189,76,126),(18,210,94,150,55,190,77,127),(19,211,95,151,56,191,78,128),(20,212,96,152,57,192,79,129),(21,213,97,153,58,193,80,130),(22,214,98,154,30,194,81,131),(23,215,99,155,31,195,82,132),(24,216,100,156,32,196,83,133),(25,217,101,157,33,197,84,134),(26,218,102,158,34,198,85,135),(27,219,103,159,35,199,86,136),(28,220,104,160,36,200,87,137),(29,221,105,161,37,201,59,138)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,58),(10,57),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40),(28,39),(29,38),(59,106),(60,105),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(117,154),(118,153),(119,152),(120,151),(121,150),(122,149),(123,148),(124,147),(125,146),(126,174),(127,173),(128,172),(129,171),(130,170),(131,169),(132,168),(133,167),(134,166),(135,165),(136,164),(137,163),(138,162),(139,161),(140,160),(141,159),(142,158),(143,157),(144,156),(145,155),(175,219),(176,218),(177,217),(178,216),(179,215),(180,214),(181,213),(182,212),(183,211),(184,210),(185,209),(186,208),(187,207),(188,206),(189,205),(190,204),(191,232),(192,231),(193,230),(194,229),(195,228),(196,227),(197,226),(198,225),(199,224),(200,223),(201,222),(202,221),(203,220)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AB | 232A | ··· | 232BD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 29 | 29 | 1 | 1 | 29 | 29 | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D29 | D58 | C4×D29 | C8×D29 |
| kernel | C8×D29 | C29⋊2C8 | C232 | C4×D29 | Dic29 | D58 | D29 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 14 | 14 | 28 | 56 |
Matrix representation of C8×D29 ►in GL2(𝔽233) generated by
| 221 | 0 |
| 0 | 221 |
| 83 | 1 |
| 232 | 0 |
| 0 | 232 |
| 232 | 0 |
G:=sub<GL(2,GF(233))| [221,0,0,221],[83,232,1,0],[0,232,232,0] >;
C8×D29 in GAP, Magma, Sage, TeX
C_8\times D_{29} % in TeX
G:=Group("C8xD29"); // GroupNames label
G:=SmallGroup(464,4);
// by ID
G=gap.SmallGroup(464,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,26,42,11204]);
// Polycyclic
G:=Group<a,b,c|a^8=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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