direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C32, C224⋊5C2, D14.2C16, C16.19D14, Dic7.2C16, C112.24C22, C7⋊C32⋊6C2, C7⋊1(C2×C32), C7⋊C8.4C8, C7⋊C16.3C4, (C4×D7).4C8, (C8×D7).5C4, C8.36(C4×D7), C2.1(D7×C16), C4.16(C8×D7), C28.21(C2×C8), C56.53(C2×C4), C14.1(C2×C16), (D7×C16).3C2, SmallGroup(448,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C32 |
Generators and relations for D7×C32
G = < a,b,c | a32=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C32
►On 224 points
Generators in S224
(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)
(1 127 212 37 152 176 95)(2 128 213 38 153 177 96)(3 97 214 39 154 178 65)(4 98 215 40 155 179 66)(5 99 216 41 156 180 67)(6 100 217 42 157 181 68)(7 101 218 43 158 182 69)(8 102 219 44 159 183 70)(9 103 220 45 160 184 71)(10 104 221 46 129 185 72)(11 105 222 47 130 186 73)(12 106 223 48 131 187 74)(13 107 224 49 132 188 75)(14 108 193 50 133 189 76)(15 109 194 51 134 190 77)(16 110 195 52 135 191 78)(17 111 196 53 136 192 79)(18 112 197 54 137 161 80)(19 113 198 55 138 162 81)(20 114 199 56 139 163 82)(21 115 200 57 140 164 83)(22 116 201 58 141 165 84)(23 117 202 59 142 166 85)(24 118 203 60 143 167 86)(25 119 204 61 144 168 87)(26 120 205 62 145 169 88)(27 121 206 63 146 170 89)(28 122 207 64 147 171 90)(29 123 208 33 148 172 91)(30 124 209 34 149 173 92)(31 125 210 35 150 174 93)(32 126 211 36 151 175 94)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(97 162)(98 163)(99 164)(100 165)(101 166)(102 167)(103 168)(104 169)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 178)(114 179)(115 180)(116 181)(117 182)(118 183)(119 184)(120 185)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 161)(129 205)(130 206)(131 207)(132 208)(133 209)(134 210)(135 211)(136 212)(137 213)(138 214)(139 215)(140 216)(141 217)(142 218)(143 219)(144 220)(145 221)(146 222)(147 223)(148 224)(149 193)(150 194)(151 195)(152 196)(153 197)(154 198)(155 199)(156 200)(157 201)(158 202)(159 203)(160 204)
G:=sub<Sym(224)| (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), (1,127,212,37,152,176,95)(2,128,213,38,153,177,96)(3,97,214,39,154,178,65)(4,98,215,40,155,179,66)(5,99,216,41,156,180,67)(6,100,217,42,157,181,68)(7,101,218,43,158,182,69)(8,102,219,44,159,183,70)(9,103,220,45,160,184,71)(10,104,221,46,129,185,72)(11,105,222,47,130,186,73)(12,106,223,48,131,187,74)(13,107,224,49,132,188,75)(14,108,193,50,133,189,76)(15,109,194,51,134,190,77)(16,110,195,52,135,191,78)(17,111,196,53,136,192,79)(18,112,197,54,137,161,80)(19,113,198,55,138,162,81)(20,114,199,56,139,163,82)(21,115,200,57,140,164,83)(22,116,201,58,141,165,84)(23,117,202,59,142,166,85)(24,118,203,60,143,167,86)(25,119,204,61,144,168,87)(26,120,205,62,145,169,88)(27,121,206,63,146,170,89)(28,122,207,64,147,171,90)(29,123,208,33,148,172,91)(30,124,209,34,149,173,92)(31,125,210,35,150,174,93)(32,126,211,36,151,175,94), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,161)(129,205)(130,206)(131,207)(132,208)(133,209)(134,210)(135,211)(136,212)(137,213)(138,214)(139,215)(140,216)(141,217)(142,218)(143,219)(144,220)(145,221)(146,222)(147,223)(148,224)(149,193)(150,194)(151,195)(152,196)(153,197)(154,198)(155,199)(156,200)(157,201)(158,202)(159,203)(160,204)>;
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), (1,127,212,37,152,176,95)(2,128,213,38,153,177,96)(3,97,214,39,154,178,65)(4,98,215,40,155,179,66)(5,99,216,41,156,180,67)(6,100,217,42,157,181,68)(7,101,218,43,158,182,69)(8,102,219,44,159,183,70)(9,103,220,45,160,184,71)(10,104,221,46,129,185,72)(11,105,222,47,130,186,73)(12,106,223,48,131,187,74)(13,107,224,49,132,188,75)(14,108,193,50,133,189,76)(15,109,194,51,134,190,77)(16,110,195,52,135,191,78)(17,111,196,53,136,192,79)(18,112,197,54,137,161,80)(19,113,198,55,138,162,81)(20,114,199,56,139,163,82)(21,115,200,57,140,164,83)(22,116,201,58,141,165,84)(23,117,202,59,142,166,85)(24,118,203,60,143,167,86)(25,119,204,61,144,168,87)(26,120,205,62,145,169,88)(27,121,206,63,146,170,89)(28,122,207,64,147,171,90)(29,123,208,33,148,172,91)(30,124,209,34,149,173,92)(31,125,210,35,150,174,93)(32,126,211,36,151,175,94), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,161)(129,205)(130,206)(131,207)(132,208)(133,209)(134,210)(135,211)(136,212)(137,213)(138,214)(139,215)(140,216)(141,217)(142,218)(143,219)(144,220)(145,221)(146,222)(147,223)(148,224)(149,193)(150,194)(151,195)(152,196)(153,197)(154,198)(155,199)(156,200)(157,201)(158,202)(159,203)(160,204) );
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)], [(1,127,212,37,152,176,95),(2,128,213,38,153,177,96),(3,97,214,39,154,178,65),(4,98,215,40,155,179,66),(5,99,216,41,156,180,67),(6,100,217,42,157,181,68),(7,101,218,43,158,182,69),(8,102,219,44,159,183,70),(9,103,220,45,160,184,71),(10,104,221,46,129,185,72),(11,105,222,47,130,186,73),(12,106,223,48,131,187,74),(13,107,224,49,132,188,75),(14,108,193,50,133,189,76),(15,109,194,51,134,190,77),(16,110,195,52,135,191,78),(17,111,196,53,136,192,79),(18,112,197,54,137,161,80),(19,113,198,55,138,162,81),(20,114,199,56,139,163,82),(21,115,200,57,140,164,83),(22,116,201,58,141,165,84),(23,117,202,59,142,166,85),(24,118,203,60,143,167,86),(25,119,204,61,144,168,87),(26,120,205,62,145,169,88),(27,121,206,63,146,170,89),(28,122,207,64,147,171,90),(29,123,208,33,148,172,91),(30,124,209,34,149,173,92),(31,125,210,35,150,174,93),(32,126,211,36,151,175,94)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(97,162),(98,163),(99,164),(100,165),(101,166),(102,167),(103,168),(104,169),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,177),(113,178),(114,179),(115,180),(116,181),(117,182),(118,183),(119,184),(120,185),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(127,192),(128,161),(129,205),(130,206),(131,207),(132,208),(133,209),(134,210),(135,211),(136,212),(137,213),(138,214),(139,215),(140,216),(141,217),(142,218),(143,219),(144,220),(145,221),(146,222),(147,223),(148,224),(149,193),(150,194),(151,195),(152,196),(153,197),(154,198),(155,199),(156,200),(157,201),(158,202),(159,203),(160,204)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 16A | ··· | 16H | 16I | ··· | 16P | 28A | ··· | 28F | 32A | ··· | 32P | 32Q | ··· | 32AF | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | D7 | D14 | C4×D7 | C8×D7 | D7×C16 | D7×C32 |
| kernel | D7×C32 | C7⋊C32 | C224 | D7×C16 | C7⋊C16 | C8×D7 | C7⋊C8 | C4×D7 | Dic7 | D14 | D7 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 3 | 3 | 6 | 12 | 24 | 48 |
Matrix representation of D7×C32 ►in GL3(𝔽449) generated by
| 128 | 0 | 0 |
| 0 | 349 | 0 |
| 0 | 0 | 349 |
| 1 | 0 | 0 |
| 0 | 398 | 2 |
| 0 | 448 | 405 |
| 1 | 0 | 0 |
| 0 | 95 | 43 |
| 0 | 354 | 354 |
G:=sub<GL(3,GF(449))| [128,0,0,0,349,0,0,0,349],[1,0,0,0,398,448,0,2,405],[1,0,0,0,95,354,0,43,354] >;
D7×C32 in GAP, Magma, Sage, TeX
D_7\times C_{32} % in TeX
G:=Group("D7xC32"); // GroupNames label
G:=SmallGroup(448,3);
// by ID
G=gap.SmallGroup(448,3);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,36,58,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^32=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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