metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6(C4×D7), C56⋊11(C2×C4), C56⋊C2⋊3C4, C56⋊C4⋊7C2, C2.D8⋊12D7, D28.9(C2×C4), (C2×C8).67D14, C14.54(C4×D4), C4⋊C4.172D14, Dic14⋊7(C2×C4), C22.92(D4×D7), Dic7⋊3Q8⋊7C2, C14.D8.7C2, D28⋊C4.7C2, C28.44(C4○D4), C14.Q16⋊23C2, C2.6(D8⋊D7), C7⋊4(SD16⋊C4), C28.51(C22×C4), C14.44(C8⋊C22), (C2×C28).305C23, (C2×C56).145C22, C4.12(Q8⋊2D7), C2.6(Q16⋊D7), (C2×Dic7).170D4, (C2×D28).85C22, C2.14(D28⋊C4), C14.73(C8.C22), (C4×Dic7).37C22, (C2×Dic14).93C22, C4.45(C2×C4×D7), (C7×C2.D8)⋊9C2, (C2×C56⋊C2).7C2, (C2×C7⋊C8).74C22, (C2×C14).310(C2×D4), (C7×C4⋊C4).98C22, (C2×C4).408(C22×D7), SmallGroup(448,423)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C2⋊C4
G = < a,b,c | a56=b2=c4=1, bab=a27, cac-1=a15, bc=cb >
Subgroups: 652 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, SD16⋊C4, C56⋊C2, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C14.D8, C14.Q16, C56⋊C4, C7×C2.D8, Dic7⋊3Q8, D28⋊C4, C2×C56⋊C2, C56⋊C2⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8⋊C22, C8.C22, C4×D7, C22×D7, SD16⋊C4, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, D8⋊D7, Q16⋊D7, C56⋊C2⋊C4
►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)
(1 162)(2 133)(3 160)(4 131)(5 158)(6 129)(7 156)(8 127)(9 154)(10 125)(11 152)(12 123)(13 150)(14 121)(15 148)(16 119)(17 146)(18 117)(19 144)(20 115)(21 142)(22 113)(23 140)(24 167)(25 138)(26 165)(27 136)(28 163)(29 134)(30 161)(31 132)(32 159)(33 130)(34 157)(35 128)(36 155)(37 126)(38 153)(39 124)(40 151)(41 122)(42 149)(43 120)(44 147)(45 118)(46 145)(47 116)(48 143)(49 114)(50 141)(51 168)(52 139)(53 166)(54 137)(55 164)(56 135)(57 185)(58 212)(59 183)(60 210)(61 181)(62 208)(63 179)(64 206)(65 177)(66 204)(67 175)(68 202)(69 173)(70 200)(71 171)(72 198)(73 169)(74 196)(75 223)(76 194)(77 221)(78 192)(79 219)(80 190)(81 217)(82 188)(83 215)(84 186)(85 213)(86 184)(87 211)(88 182)(89 209)(90 180)(91 207)(92 178)(93 205)(94 176)(95 203)(96 174)(97 201)(98 172)(99 199)(100 170)(101 197)(102 224)(103 195)(104 222)(105 193)(106 220)(107 191)(108 218)(109 189)(110 216)(111 187)(112 214)
(1 211 162 87)(2 170 163 102)(3 185 164 61)(4 200 165 76)(5 215 166 91)(6 174 167 106)(7 189 168 65)(8 204 113 80)(9 219 114 95)(10 178 115 110)(11 193 116 69)(12 208 117 84)(13 223 118 99)(14 182 119 58)(15 197 120 73)(16 212 121 88)(17 171 122 103)(18 186 123 62)(19 201 124 77)(20 216 125 92)(21 175 126 107)(22 190 127 66)(23 205 128 81)(24 220 129 96)(25 179 130 111)(26 194 131 70)(27 209 132 85)(28 224 133 100)(29 183 134 59)(30 198 135 74)(31 213 136 89)(32 172 137 104)(33 187 138 63)(34 202 139 78)(35 217 140 93)(36 176 141 108)(37 191 142 67)(38 206 143 82)(39 221 144 97)(40 180 145 112)(41 195 146 71)(42 210 147 86)(43 169 148 101)(44 184 149 60)(45 199 150 75)(46 214 151 90)(47 173 152 105)(48 188 153 64)(49 203 154 79)(50 218 155 94)(51 177 156 109)(52 192 157 68)(53 207 158 83)(54 222 159 98)(55 181 160 57)(56 196 161 72)
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), (1,162)(2,133)(3,160)(4,131)(5,158)(6,129)(7,156)(8,127)(9,154)(10,125)(11,152)(12,123)(13,150)(14,121)(15,148)(16,119)(17,146)(18,117)(19,144)(20,115)(21,142)(22,113)(23,140)(24,167)(25,138)(26,165)(27,136)(28,163)(29,134)(30,161)(31,132)(32,159)(33,130)(34,157)(35,128)(36,155)(37,126)(38,153)(39,124)(40,151)(41,122)(42,149)(43,120)(44,147)(45,118)(46,145)(47,116)(48,143)(49,114)(50,141)(51,168)(52,139)(53,166)(54,137)(55,164)(56,135)(57,185)(58,212)(59,183)(60,210)(61,181)(62,208)(63,179)(64,206)(65,177)(66,204)(67,175)(68,202)(69,173)(70,200)(71,171)(72,198)(73,169)(74,196)(75,223)(76,194)(77,221)(78,192)(79,219)(80,190)(81,217)(82,188)(83,215)(84,186)(85,213)(86,184)(87,211)(88,182)(89,209)(90,180)(91,207)(92,178)(93,205)(94,176)(95,203)(96,174)(97,201)(98,172)(99,199)(100,170)(101,197)(102,224)(103,195)(104,222)(105,193)(106,220)(107,191)(108,218)(109,189)(110,216)(111,187)(112,214), (1,211,162,87)(2,170,163,102)(3,185,164,61)(4,200,165,76)(5,215,166,91)(6,174,167,106)(7,189,168,65)(8,204,113,80)(9,219,114,95)(10,178,115,110)(11,193,116,69)(12,208,117,84)(13,223,118,99)(14,182,119,58)(15,197,120,73)(16,212,121,88)(17,171,122,103)(18,186,123,62)(19,201,124,77)(20,216,125,92)(21,175,126,107)(22,190,127,66)(23,205,128,81)(24,220,129,96)(25,179,130,111)(26,194,131,70)(27,209,132,85)(28,224,133,100)(29,183,134,59)(30,198,135,74)(31,213,136,89)(32,172,137,104)(33,187,138,63)(34,202,139,78)(35,217,140,93)(36,176,141,108)(37,191,142,67)(38,206,143,82)(39,221,144,97)(40,180,145,112)(41,195,146,71)(42,210,147,86)(43,169,148,101)(44,184,149,60)(45,199,150,75)(46,214,151,90)(47,173,152,105)(48,188,153,64)(49,203,154,79)(50,218,155,94)(51,177,156,109)(52,192,157,68)(53,207,158,83)(54,222,159,98)(55,181,160,57)(56,196,161,72)>;
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), (1,162)(2,133)(3,160)(4,131)(5,158)(6,129)(7,156)(8,127)(9,154)(10,125)(11,152)(12,123)(13,150)(14,121)(15,148)(16,119)(17,146)(18,117)(19,144)(20,115)(21,142)(22,113)(23,140)(24,167)(25,138)(26,165)(27,136)(28,163)(29,134)(30,161)(31,132)(32,159)(33,130)(34,157)(35,128)(36,155)(37,126)(38,153)(39,124)(40,151)(41,122)(42,149)(43,120)(44,147)(45,118)(46,145)(47,116)(48,143)(49,114)(50,141)(51,168)(52,139)(53,166)(54,137)(55,164)(56,135)(57,185)(58,212)(59,183)(60,210)(61,181)(62,208)(63,179)(64,206)(65,177)(66,204)(67,175)(68,202)(69,173)(70,200)(71,171)(72,198)(73,169)(74,196)(75,223)(76,194)(77,221)(78,192)(79,219)(80,190)(81,217)(82,188)(83,215)(84,186)(85,213)(86,184)(87,211)(88,182)(89,209)(90,180)(91,207)(92,178)(93,205)(94,176)(95,203)(96,174)(97,201)(98,172)(99,199)(100,170)(101,197)(102,224)(103,195)(104,222)(105,193)(106,220)(107,191)(108,218)(109,189)(110,216)(111,187)(112,214), (1,211,162,87)(2,170,163,102)(3,185,164,61)(4,200,165,76)(5,215,166,91)(6,174,167,106)(7,189,168,65)(8,204,113,80)(9,219,114,95)(10,178,115,110)(11,193,116,69)(12,208,117,84)(13,223,118,99)(14,182,119,58)(15,197,120,73)(16,212,121,88)(17,171,122,103)(18,186,123,62)(19,201,124,77)(20,216,125,92)(21,175,126,107)(22,190,127,66)(23,205,128,81)(24,220,129,96)(25,179,130,111)(26,194,131,70)(27,209,132,85)(28,224,133,100)(29,183,134,59)(30,198,135,74)(31,213,136,89)(32,172,137,104)(33,187,138,63)(34,202,139,78)(35,217,140,93)(36,176,141,108)(37,191,142,67)(38,206,143,82)(39,221,144,97)(40,180,145,112)(41,195,146,71)(42,210,147,86)(43,169,148,101)(44,184,149,60)(45,199,150,75)(46,214,151,90)(47,173,152,105)(48,188,153,64)(49,203,154,79)(50,218,155,94)(51,177,156,109)(52,192,157,68)(53,207,158,83)(54,222,159,98)(55,181,160,57)(56,196,161,72) );
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)], [(1,162),(2,133),(3,160),(4,131),(5,158),(6,129),(7,156),(8,127),(9,154),(10,125),(11,152),(12,123),(13,150),(14,121),(15,148),(16,119),(17,146),(18,117),(19,144),(20,115),(21,142),(22,113),(23,140),(24,167),(25,138),(26,165),(27,136),(28,163),(29,134),(30,161),(31,132),(32,159),(33,130),(34,157),(35,128),(36,155),(37,126),(38,153),(39,124),(40,151),(41,122),(42,149),(43,120),(44,147),(45,118),(46,145),(47,116),(48,143),(49,114),(50,141),(51,168),(52,139),(53,166),(54,137),(55,164),(56,135),(57,185),(58,212),(59,183),(60,210),(61,181),(62,208),(63,179),(64,206),(65,177),(66,204),(67,175),(68,202),(69,173),(70,200),(71,171),(72,198),(73,169),(74,196),(75,223),(76,194),(77,221),(78,192),(79,219),(80,190),(81,217),(82,188),(83,215),(84,186),(85,213),(86,184),(87,211),(88,182),(89,209),(90,180),(91,207),(92,178),(93,205),(94,176),(95,203),(96,174),(97,201),(98,172),(99,199),(100,170),(101,197),(102,224),(103,195),(104,222),(105,193),(106,220),(107,191),(108,218),(109,189),(110,216),(111,187),(112,214)], [(1,211,162,87),(2,170,163,102),(3,185,164,61),(4,200,165,76),(5,215,166,91),(6,174,167,106),(7,189,168,65),(8,204,113,80),(9,219,114,95),(10,178,115,110),(11,193,116,69),(12,208,117,84),(13,223,118,99),(14,182,119,58),(15,197,120,73),(16,212,121,88),(17,171,122,103),(18,186,123,62),(19,201,124,77),(20,216,125,92),(21,175,126,107),(22,190,127,66),(23,205,128,81),(24,220,129,96),(25,179,130,111),(26,194,131,70),(27,209,132,85),(28,224,133,100),(29,183,134,59),(30,198,135,74),(31,213,136,89),(32,172,137,104),(33,187,138,63),(34,202,139,78),(35,217,140,93),(36,176,141,108),(37,191,142,67),(38,206,143,82),(39,221,144,97),(40,180,145,112),(41,195,146,71),(42,210,147,86),(43,169,148,101),(44,184,149,60),(45,199,150,75),(46,214,151,90),(47,173,152,105),(48,188,153,64),(49,203,154,79),(50,218,155,94),(51,177,156,109),(52,192,157,68),(53,207,158,83),(54,222,159,98),(55,181,160,57),(56,196,161,72)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | C8⋊C22 | C8.C22 | Q8⋊2D7 | D4×D7 | D8⋊D7 | Q16⋊D7 |
| kernel | C56⋊C2⋊C4 | C14.D8 | C14.Q16 | C56⋊C4 | C7×C2.D8 | Dic7⋊3Q8 | D28⋊C4 | C2×C56⋊C2 | C56⋊C2 | C2×Dic7 | C2.D8 | C28 | C4⋊C4 | C2×C8 | C8 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 6 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊C2⋊C4 ►in GL8(𝔽113)
| 48 | 21 | 76 | 69 | 0 | 0 | 0 | 0 |
| 63 | 79 | 59 | 64 | 0 | 0 | 0 | 0 |
| 8 | 26 | 50 | 92 | 0 | 0 | 0 | 0 |
| 24 | 87 | 87 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 76 | 42 | 76 |
| 0 | 0 | 0 | 0 | 37 | 36 | 37 | 36 |
| 0 | 0 | 0 | 0 | 71 | 37 | 42 | 76 |
| 0 | 0 | 0 | 0 | 76 | 77 | 37 | 36 |
| 15 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 66 | 98 | 0 | 0 | 0 | 0 | 0 | 0 |
| 74 | 111 | 112 | 0 | 0 | 0 | 0 | 0 |
| 77 | 87 | 89 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 33 | 104 |
| 0 | 0 | 0 | 0 | 0 | 0 | 33 | 80 |
| 29 | 28 | 0 | 110 | 0 | 0 | 0 | 0 |
| 25 | 82 | 3 | 30 | 0 | 0 | 0 | 0 |
| 37 | 47 | 90 | 85 | 0 | 0 | 0 | 0 |
| 62 | 19 | 28 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 51 | 49 |
| 0 | 0 | 0 | 0 | 0 | 5 | 64 | 62 |
| 0 | 0 | 0 | 0 | 51 | 49 | 108 | 0 |
| 0 | 0 | 0 | 0 | 64 | 62 | 0 | 108 |
G:=sub<GL(8,GF(113))| [48,63,8,24,0,0,0,0,21,79,26,87,0,0,0,0,76,59,50,87,0,0,0,0,69,64,92,49,0,0,0,0,0,0,0,0,42,37,71,76,0,0,0,0,76,36,37,77,0,0,0,0,42,37,42,37,0,0,0,0,76,36,76,36],[15,66,74,77,0,0,0,0,24,98,111,87,0,0,0,0,0,0,112,89,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,80,80,0,0,0,0,0,0,9,33,0,0,0,0,0,0,0,0,33,33,0,0,0,0,0,0,104,80],[29,25,37,62,0,0,0,0,28,82,47,19,0,0,0,0,0,3,90,28,0,0,0,0,110,30,85,25,0,0,0,0,0,0,0,0,5,0,51,64,0,0,0,0,0,5,49,62,0,0,0,0,51,64,108,0,0,0,0,0,49,62,0,108] >;
C56⋊C2⋊C4 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_2\rtimes C_4 % in TeX
G:=Group("C56:C2:C4"); // GroupNames label
G:=SmallGroup(448,423);
// by ID
G=gap.SmallGroup(448,423);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,219,58,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^27,c*a*c^-1=a^15,b*c=c*b>;
// generators/relations