direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C22⋊C8, C23⋊3C56, C24.4C28, (C22×C56)⋊5C2, (C22×C14)⋊3C8, (C22×C8)⋊1C14, C22⋊2(C2×C56), C4.69(D4×C14), (C2×C56)⋊42C22, (C2×C28).535D4, C28.474(C2×D4), C2.1(C22×C56), (C23×C14).5C4, (C23×C28).8C2, (C23×C4).7C14, C14.30(C22×C8), C23.28(C2×C28), (C22×C4).11C28, (C22×C28).19C4, C2.3(C14×M4(2)), (C2×C28).981C23, C14.47(C2×M4(2)), (C2×C14).30M4(2), C28.114(C22⋊C4), C22.9(C7×M4(2)), C22.19(C22×C28), (C22×C28).495C22, (C2×C14)⋊7(C2×C8), (C2×C8)⋊10(C2×C14), (C2×C4).59(C2×C28), (C2×C4).145(C7×D4), C2.3(C14×C22⋊C4), C4.31(C7×C22⋊C4), (C2×C28).288(C2×C4), C14.97(C2×C22⋊C4), C22.32(C7×C22⋊C4), (C2×C4).149(C22×C14), (C22×C4).135(C2×C14), (C2×C14).231(C22×C4), (C22×C14).114(C2×C4), (C2×C14).134(C22⋊C4), SmallGroup(448,814)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C22⋊C8
G = < a,b,c,d | a14=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 290 in 202 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, C22×C8, C23×C4, C56, C2×C28, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C2×C22⋊C8, C2×C56, C2×C56, C22×C28, C22×C28, C22×C28, C23×C14, C7×C22⋊C8, C22×C56, C23×C28, C14×C22⋊C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C23, C14, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C28, C2×C14, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C56, C2×C28, C7×D4, C22×C14, C2×C22⋊C8, C7×C22⋊C4, C2×C56, C7×M4(2), C22×C28, D4×C14, C7×C22⋊C8, C14×C22⋊C4, C22×C56, C14×M4(2), C14×C22⋊C8
►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)
(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 165)(30 166)(31 167)(32 168)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 113)(54 114)(55 115)(56 116)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)
(1 191)(2 192)(3 193)(4 194)(5 195)(6 196)(7 183)(8 184)(9 185)(10 186)(11 187)(12 188)(13 189)(14 190)(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 165)(30 166)(31 167)(32 168)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 113)(54 114)(55 115)(56 116)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 141)(96 142)(97 143)(98 144)(127 222)(128 223)(129 224)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 217)(137 218)(138 219)(139 220)(140 221)(169 205)(170 206)(171 207)(172 208)(173 209)(174 210)(175 197)(176 198)(177 199)(178 200)(179 201)(180 202)(181 203)(182 204)
(1 59 142 109 132 54 209 164)(2 60 143 110 133 55 210 165)(3 61 144 111 134 56 197 166)(4 62 145 112 135 43 198 167)(5 63 146 99 136 44 199 168)(6 64 147 100 137 45 200 155)(7 65 148 101 138 46 201 156)(8 66 149 102 139 47 202 157)(9 67 150 103 140 48 203 158)(10 68 151 104 127 49 204 159)(11 69 152 105 128 50 205 160)(12 70 153 106 129 51 206 161)(13 57 154 107 130 52 207 162)(14 58 141 108 131 53 208 163)(15 216 117 176 31 194 82 85)(16 217 118 177 32 195 83 86)(17 218 119 178 33 196 84 87)(18 219 120 179 34 183 71 88)(19 220 121 180 35 184 72 89)(20 221 122 181 36 185 73 90)(21 222 123 182 37 186 74 91)(22 223 124 169 38 187 75 92)(23 224 125 170 39 188 76 93)(24 211 126 171 40 189 77 94)(25 212 113 172 41 190 78 95)(26 213 114 173 42 191 79 96)(27 214 115 174 29 192 80 97)(28 215 116 175 30 193 81 98)
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), (15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,165)(30,166)(31,167)(32,168)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,113)(54,114)(55,115)(56,116)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76), (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,165)(30,166)(31,167)(32,168)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,113)(54,114)(55,115)(56,116)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(127,222)(128,223)(129,224)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,221)(169,205)(170,206)(171,207)(172,208)(173,209)(174,210)(175,197)(176,198)(177,199)(178,200)(179,201)(180,202)(181,203)(182,204), (1,59,142,109,132,54,209,164)(2,60,143,110,133,55,210,165)(3,61,144,111,134,56,197,166)(4,62,145,112,135,43,198,167)(5,63,146,99,136,44,199,168)(6,64,147,100,137,45,200,155)(7,65,148,101,138,46,201,156)(8,66,149,102,139,47,202,157)(9,67,150,103,140,48,203,158)(10,68,151,104,127,49,204,159)(11,69,152,105,128,50,205,160)(12,70,153,106,129,51,206,161)(13,57,154,107,130,52,207,162)(14,58,141,108,131,53,208,163)(15,216,117,176,31,194,82,85)(16,217,118,177,32,195,83,86)(17,218,119,178,33,196,84,87)(18,219,120,179,34,183,71,88)(19,220,121,180,35,184,72,89)(20,221,122,181,36,185,73,90)(21,222,123,182,37,186,74,91)(22,223,124,169,38,187,75,92)(23,224,125,170,39,188,76,93)(24,211,126,171,40,189,77,94)(25,212,113,172,41,190,78,95)(26,213,114,173,42,191,79,96)(27,214,115,174,29,192,80,97)(28,215,116,175,30,193,81,98)>;
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), (15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,165)(30,166)(31,167)(32,168)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,113)(54,114)(55,115)(56,116)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76), (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,165)(30,166)(31,167)(32,168)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,113)(54,114)(55,115)(56,116)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(127,222)(128,223)(129,224)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,221)(169,205)(170,206)(171,207)(172,208)(173,209)(174,210)(175,197)(176,198)(177,199)(178,200)(179,201)(180,202)(181,203)(182,204), (1,59,142,109,132,54,209,164)(2,60,143,110,133,55,210,165)(3,61,144,111,134,56,197,166)(4,62,145,112,135,43,198,167)(5,63,146,99,136,44,199,168)(6,64,147,100,137,45,200,155)(7,65,148,101,138,46,201,156)(8,66,149,102,139,47,202,157)(9,67,150,103,140,48,203,158)(10,68,151,104,127,49,204,159)(11,69,152,105,128,50,205,160)(12,70,153,106,129,51,206,161)(13,57,154,107,130,52,207,162)(14,58,141,108,131,53,208,163)(15,216,117,176,31,194,82,85)(16,217,118,177,32,195,83,86)(17,218,119,178,33,196,84,87)(18,219,120,179,34,183,71,88)(19,220,121,180,35,184,72,89)(20,221,122,181,36,185,73,90)(21,222,123,182,37,186,74,91)(22,223,124,169,38,187,75,92)(23,224,125,170,39,188,76,93)(24,211,126,171,40,189,77,94)(25,212,113,172,41,190,78,95)(26,213,114,173,42,191,79,96)(27,214,115,174,29,192,80,97)(28,215,116,175,30,193,81,98) );
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)], [(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(29,165),(30,166),(31,167),(32,168),(33,155),(34,156),(35,157),(36,158),(37,159),(38,160),(39,161),(40,162),(41,163),(42,164),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,113),(54,114),(55,115),(56,116),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76)], [(1,191),(2,192),(3,193),(4,194),(5,195),(6,196),(7,183),(8,184),(9,185),(10,186),(11,187),(12,188),(13,189),(14,190),(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(29,165),(30,166),(31,167),(32,168),(33,155),(34,156),(35,157),(36,158),(37,159),(38,160),(39,161),(40,162),(41,163),(42,164),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,113),(54,114),(55,115),(56,116),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,141),(96,142),(97,143),(98,144),(127,222),(128,223),(129,224),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(136,217),(137,218),(138,219),(139,220),(140,221),(169,205),(170,206),(171,207),(172,208),(173,209),(174,210),(175,197),(176,198),(177,199),(178,200),(179,201),(180,202),(181,203),(182,204)], [(1,59,142,109,132,54,209,164),(2,60,143,110,133,55,210,165),(3,61,144,111,134,56,197,166),(4,62,145,112,135,43,198,167),(5,63,146,99,136,44,199,168),(6,64,147,100,137,45,200,155),(7,65,148,101,138,46,201,156),(8,66,149,102,139,47,202,157),(9,67,150,103,140,48,203,158),(10,68,151,104,127,49,204,159),(11,69,152,105,128,50,205,160),(12,70,153,106,129,51,206,161),(13,57,154,107,130,52,207,162),(14,58,141,108,131,53,208,163),(15,216,117,176,31,194,82,85),(16,217,118,177,32,195,83,86),(17,218,119,178,33,196,84,87),(18,219,120,179,34,183,71,88),(19,220,121,180,35,184,72,89),(20,221,122,181,36,185,73,90),(21,222,123,182,37,186,74,91),(22,223,124,169,38,187,75,92),(23,224,125,170,39,188,76,93),(24,211,126,171,40,189,77,94),(25,212,113,172,41,190,78,95),(26,213,114,173,42,191,79,96),(27,214,115,174,29,192,80,97),(28,215,116,175,30,193,81,98)]])
280 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 7A | ··· | 7F | 8A | ··· | 8P | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 28A | ··· | 28AV | 28AW | ··· | 28BT | 56A | ··· | 56CR |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C28 | C28 | C56 | D4 | M4(2) | C7×D4 | C7×M4(2) |
| kernel | C14×C22⋊C8 | C7×C22⋊C8 | C22×C56 | C23×C28 | C22×C28 | C23×C14 | C2×C22⋊C8 | C22×C14 | C22⋊C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 6 | 16 | 24 | 12 | 6 | 36 | 12 | 96 | 4 | 4 | 24 | 24 |
Matrix representation of C14×C22⋊C8 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 39 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 69 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 98 |
| 0 | 0 | 83 | 103 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,109,0,0,0,0,109],[112,0,0,0,0,112,0,0,0,0,1,39,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[69,0,0,0,0,1,0,0,0,0,10,83,0,0,98,103] >;
C14×C22⋊C8 in GAP, Magma, Sage, TeX
C_{14}\times C_2^2\rtimes C_8 % in TeX
G:=Group("C14xC2^2:C8"); // GroupNames label
G:=SmallGroup(448,814);
// by ID
G=gap.SmallGroup(448,814);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,124]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations