metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C32⋊3D7, C224⋊6C2, D14.C16, C7⋊1M6(2), Dic7.C16, C16.20D14, C112.25C22, C7⋊C32⋊4C2, C7⋊C8.2C8, C7⋊C16.2C4, (C4×D7).2C8, (C8×D7).2C4, C2.3(D7×C16), C4.17(C8×D7), C8.37(C4×D7), C28.22(C2×C8), C14.2(C2×C16), C56.54(C2×C4), (D7×C16).2C2, SmallGroup(448,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊D7
G = < a,b,c | a32=b7=c2=1, ab=ba, cac=a17, cbc=b-1 >
Smallest permutation representation of C32⋊D7
►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)
(1 101 178 81 64 220 153)(2 102 179 82 33 221 154)(3 103 180 83 34 222 155)(4 104 181 84 35 223 156)(5 105 182 85 36 224 157)(6 106 183 86 37 193 158)(7 107 184 87 38 194 159)(8 108 185 88 39 195 160)(9 109 186 89 40 196 129)(10 110 187 90 41 197 130)(11 111 188 91 42 198 131)(12 112 189 92 43 199 132)(13 113 190 93 44 200 133)(14 114 191 94 45 201 134)(15 115 192 95 46 202 135)(16 116 161 96 47 203 136)(17 117 162 65 48 204 137)(18 118 163 66 49 205 138)(19 119 164 67 50 206 139)(20 120 165 68 51 207 140)(21 121 166 69 52 208 141)(22 122 167 70 53 209 142)(23 123 168 71 54 210 143)(24 124 169 72 55 211 144)(25 125 170 73 56 212 145)(26 126 171 74 57 213 146)(27 127 172 75 58 214 147)(28 128 173 76 59 215 148)(29 97 174 77 60 216 149)(30 98 175 78 61 217 150)(31 99 176 79 62 218 151)(32 100 177 80 63 219 152)
(1 153)(2 138)(3 155)(4 140)(5 157)(6 142)(7 159)(8 144)(9 129)(10 146)(11 131)(12 148)(13 133)(14 150)(15 135)(16 152)(17 137)(18 154)(19 139)(20 156)(21 141)(22 158)(23 143)(24 160)(25 145)(26 130)(27 147)(28 132)(29 149)(30 134)(31 151)(32 136)(33 163)(34 180)(35 165)(36 182)(37 167)(38 184)(39 169)(40 186)(41 171)(42 188)(43 173)(44 190)(45 175)(46 192)(47 177)(48 162)(49 179)(50 164)(51 181)(52 166)(53 183)(54 168)(55 185)(56 170)(57 187)(58 172)(59 189)(60 174)(61 191)(62 176)(63 161)(64 178)(66 82)(68 84)(70 86)(72 88)(74 90)(76 92)(78 94)(80 96)(97 216)(98 201)(99 218)(100 203)(101 220)(102 205)(103 222)(104 207)(105 224)(106 209)(107 194)(108 211)(109 196)(110 213)(111 198)(112 215)(113 200)(114 217)(115 202)(116 219)(117 204)(118 221)(119 206)(120 223)(121 208)(122 193)(123 210)(124 195)(125 212)(126 197)(127 214)(128 199)
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), (1,101,178,81,64,220,153)(2,102,179,82,33,221,154)(3,103,180,83,34,222,155)(4,104,181,84,35,223,156)(5,105,182,85,36,224,157)(6,106,183,86,37,193,158)(7,107,184,87,38,194,159)(8,108,185,88,39,195,160)(9,109,186,89,40,196,129)(10,110,187,90,41,197,130)(11,111,188,91,42,198,131)(12,112,189,92,43,199,132)(13,113,190,93,44,200,133)(14,114,191,94,45,201,134)(15,115,192,95,46,202,135)(16,116,161,96,47,203,136)(17,117,162,65,48,204,137)(18,118,163,66,49,205,138)(19,119,164,67,50,206,139)(20,120,165,68,51,207,140)(21,121,166,69,52,208,141)(22,122,167,70,53,209,142)(23,123,168,71,54,210,143)(24,124,169,72,55,211,144)(25,125,170,73,56,212,145)(26,126,171,74,57,213,146)(27,127,172,75,58,214,147)(28,128,173,76,59,215,148)(29,97,174,77,60,216,149)(30,98,175,78,61,217,150)(31,99,176,79,62,218,151)(32,100,177,80,63,219,152), (1,153)(2,138)(3,155)(4,140)(5,157)(6,142)(7,159)(8,144)(9,129)(10,146)(11,131)(12,148)(13,133)(14,150)(15,135)(16,152)(17,137)(18,154)(19,139)(20,156)(21,141)(22,158)(23,143)(24,160)(25,145)(26,130)(27,147)(28,132)(29,149)(30,134)(31,151)(32,136)(33,163)(34,180)(35,165)(36,182)(37,167)(38,184)(39,169)(40,186)(41,171)(42,188)(43,173)(44,190)(45,175)(46,192)(47,177)(48,162)(49,179)(50,164)(51,181)(52,166)(53,183)(54,168)(55,185)(56,170)(57,187)(58,172)(59,189)(60,174)(61,191)(62,176)(63,161)(64,178)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)(97,216)(98,201)(99,218)(100,203)(101,220)(102,205)(103,222)(104,207)(105,224)(106,209)(107,194)(108,211)(109,196)(110,213)(111,198)(112,215)(113,200)(114,217)(115,202)(116,219)(117,204)(118,221)(119,206)(120,223)(121,208)(122,193)(123,210)(124,195)(125,212)(126,197)(127,214)(128,199)>;
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), (1,101,178,81,64,220,153)(2,102,179,82,33,221,154)(3,103,180,83,34,222,155)(4,104,181,84,35,223,156)(5,105,182,85,36,224,157)(6,106,183,86,37,193,158)(7,107,184,87,38,194,159)(8,108,185,88,39,195,160)(9,109,186,89,40,196,129)(10,110,187,90,41,197,130)(11,111,188,91,42,198,131)(12,112,189,92,43,199,132)(13,113,190,93,44,200,133)(14,114,191,94,45,201,134)(15,115,192,95,46,202,135)(16,116,161,96,47,203,136)(17,117,162,65,48,204,137)(18,118,163,66,49,205,138)(19,119,164,67,50,206,139)(20,120,165,68,51,207,140)(21,121,166,69,52,208,141)(22,122,167,70,53,209,142)(23,123,168,71,54,210,143)(24,124,169,72,55,211,144)(25,125,170,73,56,212,145)(26,126,171,74,57,213,146)(27,127,172,75,58,214,147)(28,128,173,76,59,215,148)(29,97,174,77,60,216,149)(30,98,175,78,61,217,150)(31,99,176,79,62,218,151)(32,100,177,80,63,219,152), (1,153)(2,138)(3,155)(4,140)(5,157)(6,142)(7,159)(8,144)(9,129)(10,146)(11,131)(12,148)(13,133)(14,150)(15,135)(16,152)(17,137)(18,154)(19,139)(20,156)(21,141)(22,158)(23,143)(24,160)(25,145)(26,130)(27,147)(28,132)(29,149)(30,134)(31,151)(32,136)(33,163)(34,180)(35,165)(36,182)(37,167)(38,184)(39,169)(40,186)(41,171)(42,188)(43,173)(44,190)(45,175)(46,192)(47,177)(48,162)(49,179)(50,164)(51,181)(52,166)(53,183)(54,168)(55,185)(56,170)(57,187)(58,172)(59,189)(60,174)(61,191)(62,176)(63,161)(64,178)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)(97,216)(98,201)(99,218)(100,203)(101,220)(102,205)(103,222)(104,207)(105,224)(106,209)(107,194)(108,211)(109,196)(110,213)(111,198)(112,215)(113,200)(114,217)(115,202)(116,219)(117,204)(118,221)(119,206)(120,223)(121,208)(122,193)(123,210)(124,195)(125,212)(126,197)(127,214)(128,199) );
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)], [(1,101,178,81,64,220,153),(2,102,179,82,33,221,154),(3,103,180,83,34,222,155),(4,104,181,84,35,223,156),(5,105,182,85,36,224,157),(6,106,183,86,37,193,158),(7,107,184,87,38,194,159),(8,108,185,88,39,195,160),(9,109,186,89,40,196,129),(10,110,187,90,41,197,130),(11,111,188,91,42,198,131),(12,112,189,92,43,199,132),(13,113,190,93,44,200,133),(14,114,191,94,45,201,134),(15,115,192,95,46,202,135),(16,116,161,96,47,203,136),(17,117,162,65,48,204,137),(18,118,163,66,49,205,138),(19,119,164,67,50,206,139),(20,120,165,68,51,207,140),(21,121,166,69,52,208,141),(22,122,167,70,53,209,142),(23,123,168,71,54,210,143),(24,124,169,72,55,211,144),(25,125,170,73,56,212,145),(26,126,171,74,57,213,146),(27,127,172,75,58,214,147),(28,128,173,76,59,215,148),(29,97,174,77,60,216,149),(30,98,175,78,61,217,150),(31,99,176,79,62,218,151),(32,100,177,80,63,219,152)], [(1,153),(2,138),(3,155),(4,140),(5,157),(6,142),(7,159),(8,144),(9,129),(10,146),(11,131),(12,148),(13,133),(14,150),(15,135),(16,152),(17,137),(18,154),(19,139),(20,156),(21,141),(22,158),(23,143),(24,160),(25,145),(26,130),(27,147),(28,132),(29,149),(30,134),(31,151),(32,136),(33,163),(34,180),(35,165),(36,182),(37,167),(38,184),(39,169),(40,186),(41,171),(42,188),(43,173),(44,190),(45,175),(46,192),(47,177),(48,162),(49,179),(50,164),(51,181),(52,166),(53,183),(54,168),(55,185),(56,170),(57,187),(58,172),(59,189),(60,174),(61,191),(62,176),(63,161),(64,178),(66,82),(68,84),(70,86),(72,88),(74,90),(76,92),(78,94),(80,96),(97,216),(98,201),(99,218),(100,203),(101,220),(102,205),(103,222),(104,207),(105,224),(106,209),(107,194),(108,211),(109,196),(110,213),(111,198),(112,215),(113,200),(114,217),(115,202),(116,219),(117,204),(118,221),(119,206),(120,223),(121,208),(122,193),(123,210),(124,195),(125,212),(126,197),(127,214),(128,199)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | 14B | 14C | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 28A | ··· | 28F | 32A | ··· | 32H | 32I | ··· | 32P | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 14 | 1 | 1 | 14 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | D7 | D14 | C4×D7 | M6(2) | C8×D7 | D7×C16 | C32⋊D7 |
| kernel | C32⋊D7 | C7⋊C32 | C224 | D7×C16 | C7⋊C16 | C8×D7 | C7⋊C8 | C4×D7 | Dic7 | D14 | C32 | C16 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 3 | 3 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of C32⋊D7 ►in GL2(𝔽449) generated by
| 131 | 382 |
| 67 | 318 |
| 43 | 1 |
| 448 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(449))| [131,67,382,318],[43,448,1,0],[0,1,1,0] >;
C32⋊D7 in GAP, Magma, Sage, TeX
C_{32}\rtimes D_7 % in TeX
G:=Group("C32:D7"); // GroupNames label
G:=SmallGroup(448,4);
// by ID
G=gap.SmallGroup(448,4);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,36,58,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^32=b^7=c^2=1,a*b=b*a,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations
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