metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.1D7, C14.2C42, C14.4M4(2), C7⋊C8⋊3C4, C7⋊1(C8⋊C4), (C4×C28).7C2, (C2×C28).3C4, C4.19(C4×D7), C28.24(C2×C4), (C2×C4).88D14, (C2×C4).2Dic7, C2.3(C4×Dic7), C2.1(C4.Dic7), C22.7(C2×Dic7), (C2×C28).102C22, (C2×C7⋊C8).7C2, (C2×C14).25(C2×C4), SmallGroup(224,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.D7
G = < a,b,c,d | a4=b4=c7=1, d2=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C42.D7
►Regular action on 224 points
Generators in S224
(1 153 105 172)(2 158 106 169)(3 155 107 174)(4 160 108 171)(5 157 109 176)(6 154 110 173)(7 159 111 170)(8 156 112 175)(9 36 200 56)(10 33 193 53)(11 38 194 50)(12 35 195 55)(13 40 196 52)(14 37 197 49)(15 34 198 54)(16 39 199 51)(17 31 187 89)(18 28 188 94)(19 25 189 91)(20 30 190 96)(21 27 191 93)(22 32 192 90)(23 29 185 95)(24 26 186 92)(41 115 71 182)(42 120 72 179)(43 117 65 184)(44 114 66 181)(45 119 67 178)(46 116 68 183)(47 113 69 180)(48 118 70 177)(57 82 224 212)(58 87 217 209)(59 84 218 214)(60 81 219 211)(61 86 220 216)(62 83 221 213)(63 88 222 210)(64 85 223 215)(73 133 167 121)(74 130 168 126)(75 135 161 123)(76 132 162 128)(77 129 163 125)(78 134 164 122)(79 131 165 127)(80 136 166 124)(97 205 141 149)(98 202 142 146)(99 207 143 151)(100 204 144 148)(101 201 137 145)(102 206 138 150)(103 203 139 147)(104 208 140 152)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 163 165 167)(162 164 166 168)(169 171 173 175)(170 172 174 176)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)(193 195 197 199)(194 196 198 200)(201 203 205 207)(202 204 206 208)(209 211 213 215)(210 212 214 216)(217 219 221 223)(218 220 222 224)
(1 16 98 73 24 42 213)(2 214 43 17 74 99 9)(3 10 100 75 18 44 215)(4 216 45 19 76 101 11)(5 12 102 77 20 46 209)(6 210 47 21 78 103 13)(7 14 104 79 22 48 211)(8 212 41 23 80 97 15)(25 132 201 38 160 61 119)(26 120 62 153 39 202 133)(27 134 203 40 154 63 113)(28 114 64 155 33 204 135)(29 136 205 34 156 57 115)(30 116 58 157 35 206 129)(31 130 207 36 158 59 117)(32 118 60 159 37 208 131)(49 152 127 90 177 219 170)(50 171 220 178 91 128 145)(51 146 121 92 179 221 172)(52 173 222 180 93 122 147)(53 148 123 94 181 223 174)(54 175 224 182 95 124 149)(55 150 125 96 183 217 176)(56 169 218 184 89 126 151)(65 187 168 143 200 106 84)(66 85 107 193 144 161 188)(67 189 162 137 194 108 86)(68 87 109 195 138 163 190)(69 191 164 139 196 110 88)(70 81 111 197 140 165 192)(71 185 166 141 198 112 82)(72 83 105 199 142 167 186)
(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)
G:=sub<Sym(224)| (1,153,105,172)(2,158,106,169)(3,155,107,174)(4,160,108,171)(5,157,109,176)(6,154,110,173)(7,159,111,170)(8,156,112,175)(9,36,200,56)(10,33,193,53)(11,38,194,50)(12,35,195,55)(13,40,196,52)(14,37,197,49)(15,34,198,54)(16,39,199,51)(17,31,187,89)(18,28,188,94)(19,25,189,91)(20,30,190,96)(21,27,191,93)(22,32,192,90)(23,29,185,95)(24,26,186,92)(41,115,71,182)(42,120,72,179)(43,117,65,184)(44,114,66,181)(45,119,67,178)(46,116,68,183)(47,113,69,180)(48,118,70,177)(57,82,224,212)(58,87,217,209)(59,84,218,214)(60,81,219,211)(61,86,220,216)(62,83,221,213)(63,88,222,210)(64,85,223,215)(73,133,167,121)(74,130,168,126)(75,135,161,123)(76,132,162,128)(77,129,163,125)(78,134,164,122)(79,131,165,127)(80,136,166,124)(97,205,141,149)(98,202,142,146)(99,207,143,151)(100,204,144,148)(101,201,137,145)(102,206,138,150)(103,203,139,147)(104,208,140,152), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224), (1,16,98,73,24,42,213)(2,214,43,17,74,99,9)(3,10,100,75,18,44,215)(4,216,45,19,76,101,11)(5,12,102,77,20,46,209)(6,210,47,21,78,103,13)(7,14,104,79,22,48,211)(8,212,41,23,80,97,15)(25,132,201,38,160,61,119)(26,120,62,153,39,202,133)(27,134,203,40,154,63,113)(28,114,64,155,33,204,135)(29,136,205,34,156,57,115)(30,116,58,157,35,206,129)(31,130,207,36,158,59,117)(32,118,60,159,37,208,131)(49,152,127,90,177,219,170)(50,171,220,178,91,128,145)(51,146,121,92,179,221,172)(52,173,222,180,93,122,147)(53,148,123,94,181,223,174)(54,175,224,182,95,124,149)(55,150,125,96,183,217,176)(56,169,218,184,89,126,151)(65,187,168,143,200,106,84)(66,85,107,193,144,161,188)(67,189,162,137,194,108,86)(68,87,109,195,138,163,190)(69,191,164,139,196,110,88)(70,81,111,197,140,165,192)(71,185,166,141,198,112,82)(72,83,105,199,142,167,186), (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)>;
G:=Group( (1,153,105,172)(2,158,106,169)(3,155,107,174)(4,160,108,171)(5,157,109,176)(6,154,110,173)(7,159,111,170)(8,156,112,175)(9,36,200,56)(10,33,193,53)(11,38,194,50)(12,35,195,55)(13,40,196,52)(14,37,197,49)(15,34,198,54)(16,39,199,51)(17,31,187,89)(18,28,188,94)(19,25,189,91)(20,30,190,96)(21,27,191,93)(22,32,192,90)(23,29,185,95)(24,26,186,92)(41,115,71,182)(42,120,72,179)(43,117,65,184)(44,114,66,181)(45,119,67,178)(46,116,68,183)(47,113,69,180)(48,118,70,177)(57,82,224,212)(58,87,217,209)(59,84,218,214)(60,81,219,211)(61,86,220,216)(62,83,221,213)(63,88,222,210)(64,85,223,215)(73,133,167,121)(74,130,168,126)(75,135,161,123)(76,132,162,128)(77,129,163,125)(78,134,164,122)(79,131,165,127)(80,136,166,124)(97,205,141,149)(98,202,142,146)(99,207,143,151)(100,204,144,148)(101,201,137,145)(102,206,138,150)(103,203,139,147)(104,208,140,152), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224), (1,16,98,73,24,42,213)(2,214,43,17,74,99,9)(3,10,100,75,18,44,215)(4,216,45,19,76,101,11)(5,12,102,77,20,46,209)(6,210,47,21,78,103,13)(7,14,104,79,22,48,211)(8,212,41,23,80,97,15)(25,132,201,38,160,61,119)(26,120,62,153,39,202,133)(27,134,203,40,154,63,113)(28,114,64,155,33,204,135)(29,136,205,34,156,57,115)(30,116,58,157,35,206,129)(31,130,207,36,158,59,117)(32,118,60,159,37,208,131)(49,152,127,90,177,219,170)(50,171,220,178,91,128,145)(51,146,121,92,179,221,172)(52,173,222,180,93,122,147)(53,148,123,94,181,223,174)(54,175,224,182,95,124,149)(55,150,125,96,183,217,176)(56,169,218,184,89,126,151)(65,187,168,143,200,106,84)(66,85,107,193,144,161,188)(67,189,162,137,194,108,86)(68,87,109,195,138,163,190)(69,191,164,139,196,110,88)(70,81,111,197,140,165,192)(71,185,166,141,198,112,82)(72,83,105,199,142,167,186), (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) );
G=PermutationGroup([[(1,153,105,172),(2,158,106,169),(3,155,107,174),(4,160,108,171),(5,157,109,176),(6,154,110,173),(7,159,111,170),(8,156,112,175),(9,36,200,56),(10,33,193,53),(11,38,194,50),(12,35,195,55),(13,40,196,52),(14,37,197,49),(15,34,198,54),(16,39,199,51),(17,31,187,89),(18,28,188,94),(19,25,189,91),(20,30,190,96),(21,27,191,93),(22,32,192,90),(23,29,185,95),(24,26,186,92),(41,115,71,182),(42,120,72,179),(43,117,65,184),(44,114,66,181),(45,119,67,178),(46,116,68,183),(47,113,69,180),(48,118,70,177),(57,82,224,212),(58,87,217,209),(59,84,218,214),(60,81,219,211),(61,86,220,216),(62,83,221,213),(63,88,222,210),(64,85,223,215),(73,133,167,121),(74,130,168,126),(75,135,161,123),(76,132,162,128),(77,129,163,125),(78,134,164,122),(79,131,165,127),(80,136,166,124),(97,205,141,149),(98,202,142,146),(99,207,143,151),(100,204,144,148),(101,201,137,145),(102,206,138,150),(103,203,139,147),(104,208,140,152)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,163,165,167),(162,164,166,168),(169,171,173,175),(170,172,174,176),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192),(193,195,197,199),(194,196,198,200),(201,203,205,207),(202,204,206,208),(209,211,213,215),(210,212,214,216),(217,219,221,223),(218,220,222,224)], [(1,16,98,73,24,42,213),(2,214,43,17,74,99,9),(3,10,100,75,18,44,215),(4,216,45,19,76,101,11),(5,12,102,77,20,46,209),(6,210,47,21,78,103,13),(7,14,104,79,22,48,211),(8,212,41,23,80,97,15),(25,132,201,38,160,61,119),(26,120,62,153,39,202,133),(27,134,203,40,154,63,113),(28,114,64,155,33,204,135),(29,136,205,34,156,57,115),(30,116,58,157,35,206,129),(31,130,207,36,158,59,117),(32,118,60,159,37,208,131),(49,152,127,90,177,219,170),(50,171,220,178,91,128,145),(51,146,121,92,179,221,172),(52,173,222,180,93,122,147),(53,148,123,94,181,223,174),(54,175,224,182,95,124,149),(55,150,125,96,183,217,176),(56,169,218,184,89,126,151),(65,187,168,143,200,106,84),(66,85,107,193,144,161,188),(67,189,162,137,194,108,86),(68,87,109,195,138,163,190),(69,191,164,139,196,110,88),(70,81,111,197,140,165,192),(71,185,166,141,198,112,82),(72,83,105,199,142,167,186)], [(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)]])
C42.D7 is a maximal subgroup of
C42.D14 C42.2D14 C42.7D14 C42.8D14 D14.C42 C42.243D14 D7×C8⋊C4 C42.182D14 Dic7.C42 M4(2).22D14 C42.27D14 D14⋊3M4(2) C4×C4.Dic7 C42.7Dic7 C28.5C42 C42.187D14 C42.47D14 C42.48D14 C42.51D14 C42.210D14 C42.56D14 C42.59D14 C42.62D14 C42.64D14 C42.65D14 C42.68D14 C42.70D14 C42.71D14 C42.72D14 C42.74D14 C42.76D14 C42.77D14 C42.80D14 C42.82D14
C42.D7 is a maximal quotient of
C42.279D14 C28.15C42 (C2×C28)⋊3C8
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | D7 | M4(2) | Dic7 | D14 | C4×D7 | C4.Dic7 |
| kernel | C42.D7 | C2×C7⋊C8 | C4×C28 | C7⋊C8 | C2×C28 | C42 | C14 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 4 | 6 | 3 | 12 | 24 |
Matrix representation of C42.D7 ►in GL3(𝔽113) generated by
| 15 | 0 | 0 |
| 0 | 22 | 108 |
| 0 | 74 | 91 |
| 1 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 15 |
| 1 | 0 | 0 |
| 0 | 34 | 1 |
| 0 | 53 | 88 |
| 1 | 0 | 0 |
| 0 | 100 | 43 |
| 0 | 70 | 13 |
G:=sub<GL(3,GF(113))| [15,0,0,0,22,74,0,108,91],[1,0,0,0,15,0,0,0,15],[1,0,0,0,34,53,0,1,88],[1,0,0,0,100,70,0,43,13] >;
C42.D7 in GAP, Magma, Sage, TeX
C_4^2.D_7
% in TeX
G:=Group("C4^2.D7"); // GroupNames label
G:=SmallGroup(224,9);
// by ID
G=gap.SmallGroup(224,9);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,217,55,86,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=1,d^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export