metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊C8⋊2C2, C28.34(C4⋊C4), (C2×C28).60Q8, C28.86(C2×Q8), (C22×C8).6D7, (C2×C28).481D4, C28.433(C2×D4), (C2×C8).291D14, C4⋊Dic7.13C4, (C22×C56).9C2, C23.32(C4×D7), C23.D7.8C4, C14.16(C8○D4), C4.51(C2×Dic14), (C2×C4).47Dic14, C4.26(Dic7⋊C4), (C2×C28).857C23, (C2×C56).351C22, (C22×C4).421D14, C7⋊3(C42.6C22), C2.17(D28.2C4), C22.16(Dic7⋊C4), (C22×C28).538C22, (C4×Dic7).186C22, C23.21D14.4C2, C14.43(C2×C4⋊C4), (C2×C4).113(C4×D7), C4.123(C2×C7⋊D4), (C2×C14).48(C4⋊C4), C22.140(C2×C4×D7), (C2×C28).226(C2×C4), (C2×C7⋊C8).203C22, C2.15(C2×Dic7⋊C4), (C2×C4.Dic7).4C2, (C2×C4).251(C7⋊D4), (C22×C14).92(C2×C4), (C2×Dic7).30(C2×C4), (C2×C4).799(C22×D7), (C2×C14).127(C22×C4), SmallGroup(448,636)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊C8⋊C2
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, dbd=bc4, cd=dc >
Subgroups: 356 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C42.6C22, C2×C7⋊C8, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C2×C56, C22×C28, Dic7⋊C8, C2×C4.Dic7, C23.21D14, C22×C56, Dic7⋊C8⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C8○D4, Dic14, C4×D7, C7⋊D4, C22×D7, C42.6C22, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, D28.2C4, C2×Dic7⋊C4, Dic7⋊C8⋊C2
►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)
(1 183 8 190)(2 196 9 189)(3 195 10 188)(4 194 11 187)(5 193 12 186)(6 192 13 185)(7 191 14 184)(15 201 22 208)(16 200 23 207)(17 199 24 206)(18 198 25 205)(19 197 26 204)(20 210 27 203)(21 209 28 202)(29 101 36 108)(30 100 37 107)(31 99 38 106)(32 112 39 105)(33 111 40 104)(34 110 41 103)(35 109 42 102)(43 219 50 212)(44 218 51 211)(45 217 52 224)(46 216 53 223)(47 215 54 222)(48 214 55 221)(49 213 56 220)(57 122 64 115)(58 121 65 114)(59 120 66 113)(60 119 67 126)(61 118 68 125)(62 117 69 124)(63 116 70 123)(71 93 78 86)(72 92 79 85)(73 91 80 98)(74 90 81 97)(75 89 82 96)(76 88 83 95)(77 87 84 94)(127 169 134 176)(128 182 135 175)(129 181 136 174)(130 180 137 173)(131 179 138 172)(132 178 139 171)(133 177 140 170)(141 157 148 164)(142 156 149 163)(143 155 150 162)(144 168 151 161)(145 167 152 160)(146 166 153 159)(147 165 154 158)
(1 135 79 198 123 222 105 157)(2 136 80 199 124 223 106 158)(3 137 81 200 125 224 107 159)(4 138 82 201 126 211 108 160)(5 139 83 202 113 212 109 161)(6 140 84 203 114 213 110 162)(7 127 71 204 115 214 111 163)(8 128 72 205 116 215 112 164)(9 129 73 206 117 216 99 165)(10 130 74 207 118 217 100 166)(11 131 75 208 119 218 101 167)(12 132 76 209 120 219 102 168)(13 133 77 210 121 220 103 155)(14 134 78 197 122 221 104 156)(15 60 51 29 152 194 179 96)(16 61 52 30 153 195 180 97)(17 62 53 31 154 196 181 98)(18 63 54 32 141 183 182 85)(19 64 55 33 142 184 169 86)(20 65 56 34 143 185 170 87)(21 66 43 35 144 186 171 88)(22 67 44 36 145 187 172 89)(23 68 45 37 146 188 173 90)(24 69 46 38 147 189 174 91)(25 70 47 39 148 190 175 92)(26 57 48 40 149 191 176 93)(27 58 49 41 150 192 177 94)(28 59 50 42 151 193 178 95)
(15 152)(16 153)(17 154)(18 141)(19 142)(20 143)(21 144)(22 145)(23 146)(24 147)(25 148)(26 149)(27 150)(28 151)(29 96)(30 97)(31 98)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 171)(44 172)(45 173)(46 174)(47 175)(48 176)(49 177)(50 178)(51 179)(52 180)(53 181)(54 182)(55 169)(56 170)(57 191)(58 192)(59 193)(60 194)(61 195)(62 196)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)
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), (1,183,8,190)(2,196,9,189)(3,195,10,188)(4,194,11,187)(5,193,12,186)(6,192,13,185)(7,191,14,184)(15,201,22,208)(16,200,23,207)(17,199,24,206)(18,198,25,205)(19,197,26,204)(20,210,27,203)(21,209,28,202)(29,101,36,108)(30,100,37,107)(31,99,38,106)(32,112,39,105)(33,111,40,104)(34,110,41,103)(35,109,42,102)(43,219,50,212)(44,218,51,211)(45,217,52,224)(46,216,53,223)(47,215,54,222)(48,214,55,221)(49,213,56,220)(57,122,64,115)(58,121,65,114)(59,120,66,113)(60,119,67,126)(61,118,68,125)(62,117,69,124)(63,116,70,123)(71,93,78,86)(72,92,79,85)(73,91,80,98)(74,90,81,97)(75,89,82,96)(76,88,83,95)(77,87,84,94)(127,169,134,176)(128,182,135,175)(129,181,136,174)(130,180,137,173)(131,179,138,172)(132,178,139,171)(133,177,140,170)(141,157,148,164)(142,156,149,163)(143,155,150,162)(144,168,151,161)(145,167,152,160)(146,166,153,159)(147,165,154,158), (1,135,79,198,123,222,105,157)(2,136,80,199,124,223,106,158)(3,137,81,200,125,224,107,159)(4,138,82,201,126,211,108,160)(5,139,83,202,113,212,109,161)(6,140,84,203,114,213,110,162)(7,127,71,204,115,214,111,163)(8,128,72,205,116,215,112,164)(9,129,73,206,117,216,99,165)(10,130,74,207,118,217,100,166)(11,131,75,208,119,218,101,167)(12,132,76,209,120,219,102,168)(13,133,77,210,121,220,103,155)(14,134,78,197,122,221,104,156)(15,60,51,29,152,194,179,96)(16,61,52,30,153,195,180,97)(17,62,53,31,154,196,181,98)(18,63,54,32,141,183,182,85)(19,64,55,33,142,184,169,86)(20,65,56,34,143,185,170,87)(21,66,43,35,144,186,171,88)(22,67,44,36,145,187,172,89)(23,68,45,37,146,188,173,90)(24,69,46,38,147,189,174,91)(25,70,47,39,148,190,175,92)(26,57,48,40,149,191,176,93)(27,58,49,41,150,192,177,94)(28,59,50,42,151,193,178,95), (15,152)(16,153)(17,154)(18,141)(19,142)(20,143)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,96)(30,97)(31,98)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,171)(44,172)(45,173)(46,174)(47,175)(48,176)(49,177)(50,178)(51,179)(52,180)(53,181)(54,182)(55,169)(56,170)(57,191)(58,192)(59,193)(60,194)(61,195)(62,196)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)>;
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), (1,183,8,190)(2,196,9,189)(3,195,10,188)(4,194,11,187)(5,193,12,186)(6,192,13,185)(7,191,14,184)(15,201,22,208)(16,200,23,207)(17,199,24,206)(18,198,25,205)(19,197,26,204)(20,210,27,203)(21,209,28,202)(29,101,36,108)(30,100,37,107)(31,99,38,106)(32,112,39,105)(33,111,40,104)(34,110,41,103)(35,109,42,102)(43,219,50,212)(44,218,51,211)(45,217,52,224)(46,216,53,223)(47,215,54,222)(48,214,55,221)(49,213,56,220)(57,122,64,115)(58,121,65,114)(59,120,66,113)(60,119,67,126)(61,118,68,125)(62,117,69,124)(63,116,70,123)(71,93,78,86)(72,92,79,85)(73,91,80,98)(74,90,81,97)(75,89,82,96)(76,88,83,95)(77,87,84,94)(127,169,134,176)(128,182,135,175)(129,181,136,174)(130,180,137,173)(131,179,138,172)(132,178,139,171)(133,177,140,170)(141,157,148,164)(142,156,149,163)(143,155,150,162)(144,168,151,161)(145,167,152,160)(146,166,153,159)(147,165,154,158), (1,135,79,198,123,222,105,157)(2,136,80,199,124,223,106,158)(3,137,81,200,125,224,107,159)(4,138,82,201,126,211,108,160)(5,139,83,202,113,212,109,161)(6,140,84,203,114,213,110,162)(7,127,71,204,115,214,111,163)(8,128,72,205,116,215,112,164)(9,129,73,206,117,216,99,165)(10,130,74,207,118,217,100,166)(11,131,75,208,119,218,101,167)(12,132,76,209,120,219,102,168)(13,133,77,210,121,220,103,155)(14,134,78,197,122,221,104,156)(15,60,51,29,152,194,179,96)(16,61,52,30,153,195,180,97)(17,62,53,31,154,196,181,98)(18,63,54,32,141,183,182,85)(19,64,55,33,142,184,169,86)(20,65,56,34,143,185,170,87)(21,66,43,35,144,186,171,88)(22,67,44,36,145,187,172,89)(23,68,45,37,146,188,173,90)(24,69,46,38,147,189,174,91)(25,70,47,39,148,190,175,92)(26,57,48,40,149,191,176,93)(27,58,49,41,150,192,177,94)(28,59,50,42,151,193,178,95), (15,152)(16,153)(17,154)(18,141)(19,142)(20,143)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,96)(30,97)(31,98)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,171)(44,172)(45,173)(46,174)(47,175)(48,176)(49,177)(50,178)(51,179)(52,180)(53,181)(54,182)(55,169)(56,170)(57,191)(58,192)(59,193)(60,194)(61,195)(62,196)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190) );
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)], [(1,183,8,190),(2,196,9,189),(3,195,10,188),(4,194,11,187),(5,193,12,186),(6,192,13,185),(7,191,14,184),(15,201,22,208),(16,200,23,207),(17,199,24,206),(18,198,25,205),(19,197,26,204),(20,210,27,203),(21,209,28,202),(29,101,36,108),(30,100,37,107),(31,99,38,106),(32,112,39,105),(33,111,40,104),(34,110,41,103),(35,109,42,102),(43,219,50,212),(44,218,51,211),(45,217,52,224),(46,216,53,223),(47,215,54,222),(48,214,55,221),(49,213,56,220),(57,122,64,115),(58,121,65,114),(59,120,66,113),(60,119,67,126),(61,118,68,125),(62,117,69,124),(63,116,70,123),(71,93,78,86),(72,92,79,85),(73,91,80,98),(74,90,81,97),(75,89,82,96),(76,88,83,95),(77,87,84,94),(127,169,134,176),(128,182,135,175),(129,181,136,174),(130,180,137,173),(131,179,138,172),(132,178,139,171),(133,177,140,170),(141,157,148,164),(142,156,149,163),(143,155,150,162),(144,168,151,161),(145,167,152,160),(146,166,153,159),(147,165,154,158)], [(1,135,79,198,123,222,105,157),(2,136,80,199,124,223,106,158),(3,137,81,200,125,224,107,159),(4,138,82,201,126,211,108,160),(5,139,83,202,113,212,109,161),(6,140,84,203,114,213,110,162),(7,127,71,204,115,214,111,163),(8,128,72,205,116,215,112,164),(9,129,73,206,117,216,99,165),(10,130,74,207,118,217,100,166),(11,131,75,208,119,218,101,167),(12,132,76,209,120,219,102,168),(13,133,77,210,121,220,103,155),(14,134,78,197,122,221,104,156),(15,60,51,29,152,194,179,96),(16,61,52,30,153,195,180,97),(17,62,53,31,154,196,181,98),(18,63,54,32,141,183,182,85),(19,64,55,33,142,184,169,86),(20,65,56,34,143,185,170,87),(21,66,43,35,144,186,171,88),(22,67,44,36,145,187,172,89),(23,68,45,37,146,188,173,90),(24,69,46,38,147,189,174,91),(25,70,47,39,148,190,175,92),(26,57,48,40,149,191,176,93),(27,58,49,41,150,192,177,94),(28,59,50,42,151,193,178,95)], [(15,152),(16,153),(17,154),(18,141),(19,142),(20,143),(21,144),(22,145),(23,146),(24,147),(25,148),(26,149),(27,150),(28,151),(29,96),(30,97),(31,98),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,171),(44,172),(45,173),(46,174),(47,175),(48,176),(49,177),(50,178),(51,179),(52,180),(53,181),(54,182),(55,169),(56,170),(57,191),(58,192),(59,193),(60,194),(61,195),(62,196),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D7 | D14 | D14 | C8○D4 | Dic14 | C4×D7 | C7⋊D4 | C4×D7 | D28.2C4 |
| kernel | Dic7⋊C8⋊C2 | Dic7⋊C8 | C2×C4.Dic7 | C23.21D14 | C22×C56 | C4⋊Dic7 | C23.D7 | C2×C28 | C2×C28 | C22×C8 | C2×C8 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 3 | 6 | 3 | 8 | 12 | 6 | 12 | 6 | 48 |
Matrix representation of Dic7⋊C8⋊C2 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 11 | 83 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 65 | 102 |
| 0 | 0 | 76 | 48 |
| 95 | 0 | 0 | 0 |
| 0 | 95 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 59 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,28,0,0,0,0,64,11,0,0,0,83],[0,112,0,0,112,0,0,0,0,0,65,76,0,0,102,48],[95,0,0,0,0,95,0,0,0,0,98,59,0,0,0,15],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1] >;
Dic7⋊C8⋊C2 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes C_8\rtimes C_2 % in TeX
G:=Group("Dic7:C8:C2"); // GroupNames label
G:=SmallGroup(448,636);
// by ID
G=gap.SmallGroup(448,636);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,422,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,d*b*d=b*c^4,c*d=d*c>;
// generators/relations