direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic29, C58⋊2C4, C2.2D58, C22.D29, C58.4C22, C29⋊3(C2×C4), (C2×C58).C2, SmallGroup(232,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C2×Dic29 |
Generators and relations for C2×Dic29
G = < a,b,c | a2=b58=1, c2=b29, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic29
►Regular action on 232 points
Generators in S232
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(117 204)(118 205)(119 206)(120 207)(121 208)(122 209)(123 210)(124 211)(125 212)(126 213)(127 214)(128 215)(129 216)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 224)(138 225)(139 226)(140 227)(141 228)(142 229)(143 230)(144 231)(145 232)(146 175)(147 176)(148 177)(149 178)(150 179)(151 180)(152 181)(153 182)(154 183)(155 184)(156 185)(157 186)(158 187)(159 188)(160 189)(161 190)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 197)(169 198)(170 199)(171 200)(172 201)(173 202)(174 203)
(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)
(1 146 30 117)(2 145 31 174)(3 144 32 173)(4 143 33 172)(5 142 34 171)(6 141 35 170)(7 140 36 169)(8 139 37 168)(9 138 38 167)(10 137 39 166)(11 136 40 165)(12 135 41 164)(13 134 42 163)(14 133 43 162)(15 132 44 161)(16 131 45 160)(17 130 46 159)(18 129 47 158)(19 128 48 157)(20 127 49 156)(21 126 50 155)(22 125 51 154)(23 124 52 153)(24 123 53 152)(25 122 54 151)(26 121 55 150)(27 120 56 149)(28 119 57 148)(29 118 58 147)(59 175 88 204)(60 232 89 203)(61 231 90 202)(62 230 91 201)(63 229 92 200)(64 228 93 199)(65 227 94 198)(66 226 95 197)(67 225 96 196)(68 224 97 195)(69 223 98 194)(70 222 99 193)(71 221 100 192)(72 220 101 191)(73 219 102 190)(74 218 103 189)(75 217 104 188)(76 216 105 187)(77 215 106 186)(78 214 107 185)(79 213 108 184)(80 212 109 183)(81 211 110 182)(82 210 111 181)(83 209 112 180)(84 208 113 179)(85 207 114 178)(86 206 115 177)(87 205 116 176)
G:=sub<Sym(232)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232)(146,175)(147,176)(148,177)(149,178)(150,179)(151,180)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,197)(169,198)(170,199)(171,200)(172,201)(173,202)(174,203), (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), (1,146,30,117)(2,145,31,174)(3,144,32,173)(4,143,33,172)(5,142,34,171)(6,141,35,170)(7,140,36,169)(8,139,37,168)(9,138,38,167)(10,137,39,166)(11,136,40,165)(12,135,41,164)(13,134,42,163)(14,133,43,162)(15,132,44,161)(16,131,45,160)(17,130,46,159)(18,129,47,158)(19,128,48,157)(20,127,49,156)(21,126,50,155)(22,125,51,154)(23,124,52,153)(24,123,53,152)(25,122,54,151)(26,121,55,150)(27,120,56,149)(28,119,57,148)(29,118,58,147)(59,175,88,204)(60,232,89,203)(61,231,90,202)(62,230,91,201)(63,229,92,200)(64,228,93,199)(65,227,94,198)(66,226,95,197)(67,225,96,196)(68,224,97,195)(69,223,98,194)(70,222,99,193)(71,221,100,192)(72,220,101,191)(73,219,102,190)(74,218,103,189)(75,217,104,188)(76,216,105,187)(77,215,106,186)(78,214,107,185)(79,213,108,184)(80,212,109,183)(81,211,110,182)(82,210,111,181)(83,209,112,180)(84,208,113,179)(85,207,114,178)(86,206,115,177)(87,205,116,176)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232)(146,175)(147,176)(148,177)(149,178)(150,179)(151,180)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,197)(169,198)(170,199)(171,200)(172,201)(173,202)(174,203), (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), (1,146,30,117)(2,145,31,174)(3,144,32,173)(4,143,33,172)(5,142,34,171)(6,141,35,170)(7,140,36,169)(8,139,37,168)(9,138,38,167)(10,137,39,166)(11,136,40,165)(12,135,41,164)(13,134,42,163)(14,133,43,162)(15,132,44,161)(16,131,45,160)(17,130,46,159)(18,129,47,158)(19,128,48,157)(20,127,49,156)(21,126,50,155)(22,125,51,154)(23,124,52,153)(24,123,53,152)(25,122,54,151)(26,121,55,150)(27,120,56,149)(28,119,57,148)(29,118,58,147)(59,175,88,204)(60,232,89,203)(61,231,90,202)(62,230,91,201)(63,229,92,200)(64,228,93,199)(65,227,94,198)(66,226,95,197)(67,225,96,196)(68,224,97,195)(69,223,98,194)(70,222,99,193)(71,221,100,192)(72,220,101,191)(73,219,102,190)(74,218,103,189)(75,217,104,188)(76,216,105,187)(77,215,106,186)(78,214,107,185)(79,213,108,184)(80,212,109,183)(81,211,110,182)(82,210,111,181)(83,209,112,180)(84,208,113,179)(85,207,114,178)(86,206,115,177)(87,205,116,176) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,105),(48,106),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,113),(56,114),(57,115),(58,116),(117,204),(118,205),(119,206),(120,207),(121,208),(122,209),(123,210),(124,211),(125,212),(126,213),(127,214),(128,215),(129,216),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,224),(138,225),(139,226),(140,227),(141,228),(142,229),(143,230),(144,231),(145,232),(146,175),(147,176),(148,177),(149,178),(150,179),(151,180),(152,181),(153,182),(154,183),(155,184),(156,185),(157,186),(158,187),(159,188),(160,189),(161,190),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,197),(169,198),(170,199),(171,200),(172,201),(173,202),(174,203)], [(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)], [(1,146,30,117),(2,145,31,174),(3,144,32,173),(4,143,33,172),(5,142,34,171),(6,141,35,170),(7,140,36,169),(8,139,37,168),(9,138,38,167),(10,137,39,166),(11,136,40,165),(12,135,41,164),(13,134,42,163),(14,133,43,162),(15,132,44,161),(16,131,45,160),(17,130,46,159),(18,129,47,158),(19,128,48,157),(20,127,49,156),(21,126,50,155),(22,125,51,154),(23,124,52,153),(24,123,53,152),(25,122,54,151),(26,121,55,150),(27,120,56,149),(28,119,57,148),(29,118,58,147),(59,175,88,204),(60,232,89,203),(61,231,90,202),(62,230,91,201),(63,229,92,200),(64,228,93,199),(65,227,94,198),(66,226,95,197),(67,225,96,196),(68,224,97,195),(69,223,98,194),(70,222,99,193),(71,221,100,192),(72,220,101,191),(73,219,102,190),(74,218,103,189),(75,217,104,188),(76,216,105,187),(77,215,106,186),(78,214,107,185),(79,213,108,184),(80,212,109,183),(81,211,110,182),(82,210,111,181),(83,209,112,180),(84,208,113,179),(85,207,114,178),(86,206,115,177),(87,205,116,176)]])
C2×Dic29 is a maximal subgroup of
C58.D4 C4⋊Dic29 D58⋊C4 C23.D29 C29⋊M4(2) C2×C4×D29 D4⋊2D29
C2×Dic29 is a maximal quotient of C4.Dic29 C4⋊Dic29 C23.D29
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 29A | ··· | 29N | 58A | ··· | 58AP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 |
| size | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D29 | Dic29 | D58 |
| kernel | C2×Dic29 | Dic29 | C2×C58 | C58 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 14 | 28 | 14 |
Matrix representation of C2×Dic29 ►in GL3(𝔽233) generated by
| 232 | 0 | 0 |
| 0 | 232 | 0 |
| 0 | 0 | 232 |
| 1 | 0 | 0 |
| 0 | 0 | 232 |
| 0 | 1 | 133 |
| 1 | 0 | 0 |
| 0 | 101 | 77 |
| 0 | 158 | 132 |
G:=sub<GL(3,GF(233))| [232,0,0,0,232,0,0,0,232],[1,0,0,0,0,1,0,232,133],[1,0,0,0,101,158,0,77,132] >;
C2×Dic29 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{29} % in TeX
G:=Group("C2xDic29"); // GroupNames label
G:=SmallGroup(232,7);
// by ID
G=gap.SmallGroup(232,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-29,16,3587]);
// Polycyclic
G:=Group<a,b,c|a^2=b^58=1,c^2=b^29,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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