direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C7×SD64, C224⋊4C2, C32⋊2C14, D16.C14, Q32⋊1C14, C56.69D4, C28.40D8, C14.16D16, C112.20C22, C4.2(C7×D8), C8.6(C7×D4), (C7×Q32)⋊5C2, C2.4(C7×D16), C16.3(C2×C14), (C7×D16).2C2, SmallGroup(448,176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×SD64
G = < a,b,c | a7=b32=c2=1, ab=ba, ac=ca, cbc=b15 >
Smallest permutation representation of C7×SD64
►On 224 points
Generators in S224
(1 129 220 45 116 178 96)(2 130 221 46 117 179 65)(3 131 222 47 118 180 66)(4 132 223 48 119 181 67)(5 133 224 49 120 182 68)(6 134 193 50 121 183 69)(7 135 194 51 122 184 70)(8 136 195 52 123 185 71)(9 137 196 53 124 186 72)(10 138 197 54 125 187 73)(11 139 198 55 126 188 74)(12 140 199 56 127 189 75)(13 141 200 57 128 190 76)(14 142 201 58 97 191 77)(15 143 202 59 98 192 78)(16 144 203 60 99 161 79)(17 145 204 61 100 162 80)(18 146 205 62 101 163 81)(19 147 206 63 102 164 82)(20 148 207 64 103 165 83)(21 149 208 33 104 166 84)(22 150 209 34 105 167 85)(23 151 210 35 106 168 86)(24 152 211 36 107 169 87)(25 153 212 37 108 170 88)(26 154 213 38 109 171 89)(27 155 214 39 110 172 90)(28 156 215 40 111 173 91)(29 157 216 41 112 174 92)(30 158 217 42 113 175 93)(31 159 218 43 114 176 94)(32 160 219 44 115 177 95)
(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 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 57)(34 40)(35 55)(36 38)(37 53)(39 51)(41 49)(42 64)(43 47)(44 62)(46 60)(48 58)(50 56)(52 54)(59 63)(65 79)(66 94)(67 77)(68 92)(69 75)(70 90)(71 73)(72 88)(74 86)(76 84)(78 82)(81 95)(83 93)(85 91)(87 89)(97 119)(98 102)(99 117)(101 115)(103 113)(104 128)(105 111)(106 126)(107 109)(108 124)(110 122)(112 120)(114 118)(121 127)(123 125)(130 144)(131 159)(132 142)(133 157)(134 140)(135 155)(136 138)(137 153)(139 151)(141 149)(143 147)(146 160)(148 158)(150 156)(152 154)(161 179)(163 177)(164 192)(165 175)(166 190)(167 173)(168 188)(169 171)(170 186)(172 184)(174 182)(176 180)(181 191)(183 189)(185 187)(193 199)(194 214)(195 197)(196 212)(198 210)(200 208)(201 223)(202 206)(203 221)(205 219)(207 217)(209 215)(211 213)(216 224)(218 222)
G:=sub<Sym(224)| (1,129,220,45,116,178,96)(2,130,221,46,117,179,65)(3,131,222,47,118,180,66)(4,132,223,48,119,181,67)(5,133,224,49,120,182,68)(6,134,193,50,121,183,69)(7,135,194,51,122,184,70)(8,136,195,52,123,185,71)(9,137,196,53,124,186,72)(10,138,197,54,125,187,73)(11,139,198,55,126,188,74)(12,140,199,56,127,189,75)(13,141,200,57,128,190,76)(14,142,201,58,97,191,77)(15,143,202,59,98,192,78)(16,144,203,60,99,161,79)(17,145,204,61,100,162,80)(18,146,205,62,101,163,81)(19,147,206,63,102,164,82)(20,148,207,64,103,165,83)(21,149,208,33,104,166,84)(22,150,209,34,105,167,85)(23,151,210,35,106,168,86)(24,152,211,36,107,169,87)(25,153,212,37,108,170,88)(26,154,213,38,109,171,89)(27,155,214,39,110,172,90)(28,156,215,40,111,173,91)(29,157,216,41,112,174,92)(30,158,217,42,113,175,93)(31,159,218,43,114,176,94)(32,160,219,44,115,177,95), (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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,57)(34,40)(35,55)(36,38)(37,53)(39,51)(41,49)(42,64)(43,47)(44,62)(46,60)(48,58)(50,56)(52,54)(59,63)(65,79)(66,94)(67,77)(68,92)(69,75)(70,90)(71,73)(72,88)(74,86)(76,84)(78,82)(81,95)(83,93)(85,91)(87,89)(97,119)(98,102)(99,117)(101,115)(103,113)(104,128)(105,111)(106,126)(107,109)(108,124)(110,122)(112,120)(114,118)(121,127)(123,125)(130,144)(131,159)(132,142)(133,157)(134,140)(135,155)(136,138)(137,153)(139,151)(141,149)(143,147)(146,160)(148,158)(150,156)(152,154)(161,179)(163,177)(164,192)(165,175)(166,190)(167,173)(168,188)(169,171)(170,186)(172,184)(174,182)(176,180)(181,191)(183,189)(185,187)(193,199)(194,214)(195,197)(196,212)(198,210)(200,208)(201,223)(202,206)(203,221)(205,219)(207,217)(209,215)(211,213)(216,224)(218,222)>;
G:=Group( (1,129,220,45,116,178,96)(2,130,221,46,117,179,65)(3,131,222,47,118,180,66)(4,132,223,48,119,181,67)(5,133,224,49,120,182,68)(6,134,193,50,121,183,69)(7,135,194,51,122,184,70)(8,136,195,52,123,185,71)(9,137,196,53,124,186,72)(10,138,197,54,125,187,73)(11,139,198,55,126,188,74)(12,140,199,56,127,189,75)(13,141,200,57,128,190,76)(14,142,201,58,97,191,77)(15,143,202,59,98,192,78)(16,144,203,60,99,161,79)(17,145,204,61,100,162,80)(18,146,205,62,101,163,81)(19,147,206,63,102,164,82)(20,148,207,64,103,165,83)(21,149,208,33,104,166,84)(22,150,209,34,105,167,85)(23,151,210,35,106,168,86)(24,152,211,36,107,169,87)(25,153,212,37,108,170,88)(26,154,213,38,109,171,89)(27,155,214,39,110,172,90)(28,156,215,40,111,173,91)(29,157,216,41,112,174,92)(30,158,217,42,113,175,93)(31,159,218,43,114,176,94)(32,160,219,44,115,177,95), (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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,57)(34,40)(35,55)(36,38)(37,53)(39,51)(41,49)(42,64)(43,47)(44,62)(46,60)(48,58)(50,56)(52,54)(59,63)(65,79)(66,94)(67,77)(68,92)(69,75)(70,90)(71,73)(72,88)(74,86)(76,84)(78,82)(81,95)(83,93)(85,91)(87,89)(97,119)(98,102)(99,117)(101,115)(103,113)(104,128)(105,111)(106,126)(107,109)(108,124)(110,122)(112,120)(114,118)(121,127)(123,125)(130,144)(131,159)(132,142)(133,157)(134,140)(135,155)(136,138)(137,153)(139,151)(141,149)(143,147)(146,160)(148,158)(150,156)(152,154)(161,179)(163,177)(164,192)(165,175)(166,190)(167,173)(168,188)(169,171)(170,186)(172,184)(174,182)(176,180)(181,191)(183,189)(185,187)(193,199)(194,214)(195,197)(196,212)(198,210)(200,208)(201,223)(202,206)(203,221)(205,219)(207,217)(209,215)(211,213)(216,224)(218,222) );
G=PermutationGroup([[(1,129,220,45,116,178,96),(2,130,221,46,117,179,65),(3,131,222,47,118,180,66),(4,132,223,48,119,181,67),(5,133,224,49,120,182,68),(6,134,193,50,121,183,69),(7,135,194,51,122,184,70),(8,136,195,52,123,185,71),(9,137,196,53,124,186,72),(10,138,197,54,125,187,73),(11,139,198,55,126,188,74),(12,140,199,56,127,189,75),(13,141,200,57,128,190,76),(14,142,201,58,97,191,77),(15,143,202,59,98,192,78),(16,144,203,60,99,161,79),(17,145,204,61,100,162,80),(18,146,205,62,101,163,81),(19,147,206,63,102,164,82),(20,148,207,64,103,165,83),(21,149,208,33,104,166,84),(22,150,209,34,105,167,85),(23,151,210,35,106,168,86),(24,152,211,36,107,169,87),(25,153,212,37,108,170,88),(26,154,213,38,109,171,89),(27,155,214,39,110,172,90),(28,156,215,40,111,173,91),(29,157,216,41,112,174,92),(30,158,217,42,113,175,93),(31,159,218,43,114,176,94),(32,160,219,44,115,177,95)], [(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)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26),(33,57),(34,40),(35,55),(36,38),(37,53),(39,51),(41,49),(42,64),(43,47),(44,62),(46,60),(48,58),(50,56),(52,54),(59,63),(65,79),(66,94),(67,77),(68,92),(69,75),(70,90),(71,73),(72,88),(74,86),(76,84),(78,82),(81,95),(83,93),(85,91),(87,89),(97,119),(98,102),(99,117),(101,115),(103,113),(104,128),(105,111),(106,126),(107,109),(108,124),(110,122),(112,120),(114,118),(121,127),(123,125),(130,144),(131,159),(132,142),(133,157),(134,140),(135,155),(136,138),(137,153),(139,151),(141,149),(143,147),(146,160),(148,158),(150,156),(152,154),(161,179),(163,177),(164,192),(165,175),(166,190),(167,173),(168,188),(169,171),(170,186),(172,184),(174,182),(176,180),(181,191),(183,189),(185,187),(193,199),(194,214),(195,197),(196,212),(198,210),(200,208),(201,223),(202,206),(203,221),(205,219),(207,217),(209,215),(211,213),(216,224),(218,222)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | ··· | 28L | 32A | ··· | 32H | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 16 | 2 | 16 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 16 | ··· | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 16 | ··· | 16 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D8 | D16 | C7×D4 | SD64 | C7×D8 | C7×D16 | C7×SD64 |
| kernel | C7×SD64 | C224 | C7×D16 | C7×Q32 | SD64 | C32 | D16 | Q32 | C56 | C28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 2 | 4 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of C7×SD64 ►in GL2(𝔽449) generated by
| 25 | 0 |
| 0 | 25 |
| 271 | 242 |
| 207 | 271 |
| 1 | 0 |
| 0 | 448 |
G:=sub<GL(2,GF(449))| [25,0,0,25],[271,207,242,271],[1,0,0,448] >;
C7×SD64 in GAP, Magma, Sage, TeX
C_7\times {\rm SD}_{64} % in TeX
G:=Group("C7xSD64"); // GroupNames label
G:=SmallGroup(448,176);
// by ID
G=gap.SmallGroup(448,176);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,1568,421,2355,1186,192,5884,2951,242,14117,7068,124]);
// 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^15>;
// generators/relations
Export