direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×Q32, C16.C14, Q16.C14, C112.2C2, C14.17D8, C28.38D4, C56.26C22, C4.3(C7×D4), C2.5(C7×D8), C8.4(C2×C14), (C7×Q16).2C2, SmallGroup(224,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q32
G = < a,b,c | a7=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Q32
►Regular action on 224 points
Generators in S224
(1 77 197 155 46 115 176)(2 78 198 156 47 116 161)(3 79 199 157 48 117 162)(4 80 200 158 33 118 163)(5 65 201 159 34 119 164)(6 66 202 160 35 120 165)(7 67 203 145 36 121 166)(8 68 204 146 37 122 167)(9 69 205 147 38 123 168)(10 70 206 148 39 124 169)(11 71 207 149 40 125 170)(12 72 208 150 41 126 171)(13 73 193 151 42 127 172)(14 74 194 152 43 128 173)(15 75 195 153 44 113 174)(16 76 196 154 45 114 175)(17 217 187 88 58 106 134)(18 218 188 89 59 107 135)(19 219 189 90 60 108 136)(20 220 190 91 61 109 137)(21 221 191 92 62 110 138)(22 222 192 93 63 111 139)(23 223 177 94 64 112 140)(24 224 178 95 49 97 141)(25 209 179 96 50 98 142)(26 210 180 81 51 99 143)(27 211 181 82 52 100 144)(28 212 182 83 53 101 129)(29 213 183 84 54 102 130)(30 214 184 85 55 103 131)(31 215 185 86 56 104 132)(32 216 186 87 57 105 133)
(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)
(1 136 9 144)(2 135 10 143)(3 134 11 142)(4 133 12 141)(5 132 13 140)(6 131 14 139)(7 130 15 138)(8 129 16 137)(17 71 25 79)(18 70 26 78)(19 69 27 77)(20 68 28 76)(21 67 29 75)(22 66 30 74)(23 65 31 73)(24 80 32 72)(33 87 41 95)(34 86 42 94)(35 85 43 93)(36 84 44 92)(37 83 45 91)(38 82 46 90)(39 81 47 89)(40 96 48 88)(49 118 57 126)(50 117 58 125)(51 116 59 124)(52 115 60 123)(53 114 61 122)(54 113 62 121)(55 128 63 120)(56 127 64 119)(97 163 105 171)(98 162 106 170)(99 161 107 169)(100 176 108 168)(101 175 109 167)(102 174 110 166)(103 173 111 165)(104 172 112 164)(145 183 153 191)(146 182 154 190)(147 181 155 189)(148 180 156 188)(149 179 157 187)(150 178 158 186)(151 177 159 185)(152 192 160 184)(193 223 201 215)(194 222 202 214)(195 221 203 213)(196 220 204 212)(197 219 205 211)(198 218 206 210)(199 217 207 209)(200 216 208 224)
G:=sub<Sym(224)| (1,77,197,155,46,115,176)(2,78,198,156,47,116,161)(3,79,199,157,48,117,162)(4,80,200,158,33,118,163)(5,65,201,159,34,119,164)(6,66,202,160,35,120,165)(7,67,203,145,36,121,166)(8,68,204,146,37,122,167)(9,69,205,147,38,123,168)(10,70,206,148,39,124,169)(11,71,207,149,40,125,170)(12,72,208,150,41,126,171)(13,73,193,151,42,127,172)(14,74,194,152,43,128,173)(15,75,195,153,44,113,174)(16,76,196,154,45,114,175)(17,217,187,88,58,106,134)(18,218,188,89,59,107,135)(19,219,189,90,60,108,136)(20,220,190,91,61,109,137)(21,221,191,92,62,110,138)(22,222,192,93,63,111,139)(23,223,177,94,64,112,140)(24,224,178,95,49,97,141)(25,209,179,96,50,98,142)(26,210,180,81,51,99,143)(27,211,181,82,52,100,144)(28,212,182,83,53,101,129)(29,213,183,84,54,102,130)(30,214,184,85,55,103,131)(31,215,185,86,56,104,132)(32,216,186,87,57,105,133), (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), (1,136,9,144)(2,135,10,143)(3,134,11,142)(4,133,12,141)(5,132,13,140)(6,131,14,139)(7,130,15,138)(8,129,16,137)(17,71,25,79)(18,70,26,78)(19,69,27,77)(20,68,28,76)(21,67,29,75)(22,66,30,74)(23,65,31,73)(24,80,32,72)(33,87,41,95)(34,86,42,94)(35,85,43,93)(36,84,44,92)(37,83,45,91)(38,82,46,90)(39,81,47,89)(40,96,48,88)(49,118,57,126)(50,117,58,125)(51,116,59,124)(52,115,60,123)(53,114,61,122)(54,113,62,121)(55,128,63,120)(56,127,64,119)(97,163,105,171)(98,162,106,170)(99,161,107,169)(100,176,108,168)(101,175,109,167)(102,174,110,166)(103,173,111,165)(104,172,112,164)(145,183,153,191)(146,182,154,190)(147,181,155,189)(148,180,156,188)(149,179,157,187)(150,178,158,186)(151,177,159,185)(152,192,160,184)(193,223,201,215)(194,222,202,214)(195,221,203,213)(196,220,204,212)(197,219,205,211)(198,218,206,210)(199,217,207,209)(200,216,208,224)>;
G:=Group( (1,77,197,155,46,115,176)(2,78,198,156,47,116,161)(3,79,199,157,48,117,162)(4,80,200,158,33,118,163)(5,65,201,159,34,119,164)(6,66,202,160,35,120,165)(7,67,203,145,36,121,166)(8,68,204,146,37,122,167)(9,69,205,147,38,123,168)(10,70,206,148,39,124,169)(11,71,207,149,40,125,170)(12,72,208,150,41,126,171)(13,73,193,151,42,127,172)(14,74,194,152,43,128,173)(15,75,195,153,44,113,174)(16,76,196,154,45,114,175)(17,217,187,88,58,106,134)(18,218,188,89,59,107,135)(19,219,189,90,60,108,136)(20,220,190,91,61,109,137)(21,221,191,92,62,110,138)(22,222,192,93,63,111,139)(23,223,177,94,64,112,140)(24,224,178,95,49,97,141)(25,209,179,96,50,98,142)(26,210,180,81,51,99,143)(27,211,181,82,52,100,144)(28,212,182,83,53,101,129)(29,213,183,84,54,102,130)(30,214,184,85,55,103,131)(31,215,185,86,56,104,132)(32,216,186,87,57,105,133), (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), (1,136,9,144)(2,135,10,143)(3,134,11,142)(4,133,12,141)(5,132,13,140)(6,131,14,139)(7,130,15,138)(8,129,16,137)(17,71,25,79)(18,70,26,78)(19,69,27,77)(20,68,28,76)(21,67,29,75)(22,66,30,74)(23,65,31,73)(24,80,32,72)(33,87,41,95)(34,86,42,94)(35,85,43,93)(36,84,44,92)(37,83,45,91)(38,82,46,90)(39,81,47,89)(40,96,48,88)(49,118,57,126)(50,117,58,125)(51,116,59,124)(52,115,60,123)(53,114,61,122)(54,113,62,121)(55,128,63,120)(56,127,64,119)(97,163,105,171)(98,162,106,170)(99,161,107,169)(100,176,108,168)(101,175,109,167)(102,174,110,166)(103,173,111,165)(104,172,112,164)(145,183,153,191)(146,182,154,190)(147,181,155,189)(148,180,156,188)(149,179,157,187)(150,178,158,186)(151,177,159,185)(152,192,160,184)(193,223,201,215)(194,222,202,214)(195,221,203,213)(196,220,204,212)(197,219,205,211)(198,218,206,210)(199,217,207,209)(200,216,208,224) );
G=PermutationGroup([[(1,77,197,155,46,115,176),(2,78,198,156,47,116,161),(3,79,199,157,48,117,162),(4,80,200,158,33,118,163),(5,65,201,159,34,119,164),(6,66,202,160,35,120,165),(7,67,203,145,36,121,166),(8,68,204,146,37,122,167),(9,69,205,147,38,123,168),(10,70,206,148,39,124,169),(11,71,207,149,40,125,170),(12,72,208,150,41,126,171),(13,73,193,151,42,127,172),(14,74,194,152,43,128,173),(15,75,195,153,44,113,174),(16,76,196,154,45,114,175),(17,217,187,88,58,106,134),(18,218,188,89,59,107,135),(19,219,189,90,60,108,136),(20,220,190,91,61,109,137),(21,221,191,92,62,110,138),(22,222,192,93,63,111,139),(23,223,177,94,64,112,140),(24,224,178,95,49,97,141),(25,209,179,96,50,98,142),(26,210,180,81,51,99,143),(27,211,181,82,52,100,144),(28,212,182,83,53,101,129),(29,213,183,84,54,102,130),(30,214,184,85,55,103,131),(31,215,185,86,56,104,132),(32,216,186,87,57,105,133)], [(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)], [(1,136,9,144),(2,135,10,143),(3,134,11,142),(4,133,12,141),(5,132,13,140),(6,131,14,139),(7,130,15,138),(8,129,16,137),(17,71,25,79),(18,70,26,78),(19,69,27,77),(20,68,28,76),(21,67,29,75),(22,66,30,74),(23,65,31,73),(24,80,32,72),(33,87,41,95),(34,86,42,94),(35,85,43,93),(36,84,44,92),(37,83,45,91),(38,82,46,90),(39,81,47,89),(40,96,48,88),(49,118,57,126),(50,117,58,125),(51,116,59,124),(52,115,60,123),(53,114,61,122),(54,113,62,121),(55,128,63,120),(56,127,64,119),(97,163,105,171),(98,162,106,170),(99,161,107,169),(100,176,108,168),(101,175,109,167),(102,174,110,166),(103,173,111,165),(104,172,112,164),(145,183,153,191),(146,182,154,190),(147,181,155,189),(148,180,156,188),(149,179,157,187),(150,178,158,186),(151,177,159,185),(152,192,160,184),(193,223,201,215),(194,222,202,214),(195,221,203,213),(196,220,204,212),(197,219,205,211),(198,218,206,210),(199,217,207,209),(200,216,208,224)]])
C7×Q32 is a maximal subgroup of
C7⋊SD64 C7⋊Q64 Q32⋊D7 Q32⋊3D7
77 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L | 112A | ··· | 112X |
| order | 1 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | D8 | Q32 | C7×D4 | C7×D8 | C7×Q32 |
| kernel | C7×Q32 | C112 | C7×Q16 | Q32 | C16 | Q16 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 4 | 6 | 12 | 24 |
Matrix representation of C7×Q32 ►in GL2(𝔽113) generated by
| 49 | 0 |
| 0 | 49 |
| 109 | 95 |
| 18 | 109 |
| 102 | 45 |
| 45 | 11 |
G:=sub<GL(2,GF(113))| [49,0,0,49],[109,18,95,109],[102,45,45,11] >;
C7×Q32 in GAP, Magma, Sage, TeX
C_7\times Q_{32} % in TeX
G:=Group("C7xQ32"); // GroupNames label
G:=SmallGroup(224,62);
// by ID
G=gap.SmallGroup(224,62);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,672,361,679,2019,1017,165,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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