direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C7×D32, C32⋊1C14, C224⋊3C2, D16⋊1C14, C28.39D8, C56.68D4, C14.15D16, C112.19C22, C8.5(C7×D4), C4.1(C7×D8), (C7×D16)⋊5C2, C2.3(C7×D16), C16.2(C2×C14), SmallGroup(448,175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D32
G = < a,b,c | a7=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D32
►On 224 points
Generators in S224
(1 134 197 162 107 57 79)(2 135 198 163 108 58 80)(3 136 199 164 109 59 81)(4 137 200 165 110 60 82)(5 138 201 166 111 61 83)(6 139 202 167 112 62 84)(7 140 203 168 113 63 85)(8 141 204 169 114 64 86)(9 142 205 170 115 33 87)(10 143 206 171 116 34 88)(11 144 207 172 117 35 89)(12 145 208 173 118 36 90)(13 146 209 174 119 37 91)(14 147 210 175 120 38 92)(15 148 211 176 121 39 93)(16 149 212 177 122 40 94)(17 150 213 178 123 41 95)(18 151 214 179 124 42 96)(19 152 215 180 125 43 65)(20 153 216 181 126 44 66)(21 154 217 182 127 45 67)(22 155 218 183 128 46 68)(23 156 219 184 97 47 69)(24 157 220 185 98 48 70)(25 158 221 186 99 49 71)(26 159 222 187 100 50 72)(27 160 223 188 101 51 73)(28 129 224 189 102 52 74)(29 130 193 190 103 53 75)(30 131 194 191 104 54 76)(31 132 195 192 105 55 77)(32 133 196 161 106 56 78)
(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)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(94 96)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(118 128)(119 127)(120 126)(121 125)(122 124)(129 139)(130 138)(131 137)(132 136)(133 135)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(161 163)(164 192)(165 191)(166 190)(167 189)(168 188)(169 187)(170 186)(171 185)(172 184)(173 183)(174 182)(175 181)(176 180)(177 179)(193 201)(194 200)(195 199)(196 198)(202 224)(203 223)(204 222)(205 221)(206 220)(207 219)(208 218)(209 217)(210 216)(211 215)(212 214)
G:=sub<Sym(224)| (1,134,197,162,107,57,79)(2,135,198,163,108,58,80)(3,136,199,164,109,59,81)(4,137,200,165,110,60,82)(5,138,201,166,111,61,83)(6,139,202,167,112,62,84)(7,140,203,168,113,63,85)(8,141,204,169,114,64,86)(9,142,205,170,115,33,87)(10,143,206,171,116,34,88)(11,144,207,172,117,35,89)(12,145,208,173,118,36,90)(13,146,209,174,119,37,91)(14,147,210,175,120,38,92)(15,148,211,176,121,39,93)(16,149,212,177,122,40,94)(17,150,213,178,123,41,95)(18,151,214,179,124,42,96)(19,152,215,180,125,43,65)(20,153,216,181,126,44,66)(21,154,217,182,127,45,67)(22,155,218,183,128,46,68)(23,156,219,184,97,47,69)(24,157,220,185,98,48,70)(25,158,221,186,99,49,71)(26,159,222,187,100,50,72)(27,160,223,188,101,51,73)(28,129,224,189,102,52,74)(29,130,193,190,103,53,75)(30,131,194,191,104,54,76)(31,132,195,192,105,55,77)(32,133,196,161,106,56,78), (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), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(94,96)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(118,128)(119,127)(120,126)(121,125)(122,124)(129,139)(130,138)(131,137)(132,136)(133,135)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(161,163)(164,192)(165,191)(166,190)(167,189)(168,188)(169,187)(170,186)(171,185)(172,184)(173,183)(174,182)(175,181)(176,180)(177,179)(193,201)(194,200)(195,199)(196,198)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216)(211,215)(212,214)>;
G:=Group( (1,134,197,162,107,57,79)(2,135,198,163,108,58,80)(3,136,199,164,109,59,81)(4,137,200,165,110,60,82)(5,138,201,166,111,61,83)(6,139,202,167,112,62,84)(7,140,203,168,113,63,85)(8,141,204,169,114,64,86)(9,142,205,170,115,33,87)(10,143,206,171,116,34,88)(11,144,207,172,117,35,89)(12,145,208,173,118,36,90)(13,146,209,174,119,37,91)(14,147,210,175,120,38,92)(15,148,211,176,121,39,93)(16,149,212,177,122,40,94)(17,150,213,178,123,41,95)(18,151,214,179,124,42,96)(19,152,215,180,125,43,65)(20,153,216,181,126,44,66)(21,154,217,182,127,45,67)(22,155,218,183,128,46,68)(23,156,219,184,97,47,69)(24,157,220,185,98,48,70)(25,158,221,186,99,49,71)(26,159,222,187,100,50,72)(27,160,223,188,101,51,73)(28,129,224,189,102,52,74)(29,130,193,190,103,53,75)(30,131,194,191,104,54,76)(31,132,195,192,105,55,77)(32,133,196,161,106,56,78), (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), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(94,96)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)(118,128)(119,127)(120,126)(121,125)(122,124)(129,139)(130,138)(131,137)(132,136)(133,135)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(161,163)(164,192)(165,191)(166,190)(167,189)(168,188)(169,187)(170,186)(171,185)(172,184)(173,183)(174,182)(175,181)(176,180)(177,179)(193,201)(194,200)(195,199)(196,198)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216)(211,215)(212,214) );
G=PermutationGroup([[(1,134,197,162,107,57,79),(2,135,198,163,108,58,80),(3,136,199,164,109,59,81),(4,137,200,165,110,60,82),(5,138,201,166,111,61,83),(6,139,202,167,112,62,84),(7,140,203,168,113,63,85),(8,141,204,169,114,64,86),(9,142,205,170,115,33,87),(10,143,206,171,116,34,88),(11,144,207,172,117,35,89),(12,145,208,173,118,36,90),(13,146,209,174,119,37,91),(14,147,210,175,120,38,92),(15,148,211,176,121,39,93),(16,149,212,177,122,40,94),(17,150,213,178,123,41,95),(18,151,214,179,124,42,96),(19,152,215,180,125,43,65),(20,153,216,181,126,44,66),(21,154,217,182,127,45,67),(22,155,218,183,128,46,68),(23,156,219,184,97,47,69),(24,157,220,185,98,48,70),(25,158,221,186,99,49,71),(26,159,222,187,100,50,72),(27,160,223,188,101,51,73),(28,129,224,189,102,52,74),(29,130,193,190,103,53,75),(30,131,194,191,104,54,76),(31,132,195,192,105,55,77),(32,133,196,161,106,56,78)], [(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)], [(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(94,96),(97,117),(98,116),(99,115),(100,114),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108),(118,128),(119,127),(120,126),(121,125),(122,124),(129,139),(130,138),(131,137),(132,136),(133,135),(140,160),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151),(161,163),(164,192),(165,191),(166,190),(167,189),(168,188),(169,187),(170,186),(171,185),(172,184),(173,183),(174,182),(175,181),(176,180),(177,179),(193,201),(194,200),(195,199),(196,198),(202,224),(203,223),(204,222),(205,221),(206,220),(207,219),(208,218),(209,217),(210,216),(211,215),(212,214)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14R | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 32A | ··· | 32H | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 16 | 16 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 16 | ··· | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | D8 | D16 | C7×D4 | D32 | C7×D8 | C7×D16 | C7×D32 |
| kernel | C7×D32 | C224 | C7×D16 | D32 | C32 | D16 | C56 | C28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 4 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of C7×D32 ►in GL2(𝔽449) generated by
| 324 | 0 |
| 0 | 324 |
| 409 | 113 |
| 336 | 409 |
| 1 | 0 |
| 0 | 448 |
G:=sub<GL(2,GF(449))| [324,0,0,324],[409,336,113,409],[1,0,0,448] >;
C7×D32 in GAP, Magma, Sage, TeX
C_7\times D_{32} % in TeX
G:=Group("C7xD32"); // GroupNames label
G:=SmallGroup(448,175);
// by ID
G=gap.SmallGroup(448,175);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,421,2355,1186,192,5884,2951,242,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c|a^7=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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