metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊9D4, C7⋊C8⋊10D4, C8⋊6(C7⋊D4), C7⋊5(C8⋊3D4), C56⋊C4⋊6C2, C4.26(D4×D7), (C2×D56)⋊25C2, C28⋊D4⋊6C2, (C2×SD16)⋊2D7, (C2×C8).91D14, (C14×SD16)⋊4C2, (C2×D4).77D14, C28.179(C2×D4), (C2×Q8).58D14, C28.23D4⋊5C2, (C2×Dic7).74D4, C22.273(D4×D7), C2.23(C28⋊D4), C2.31(D56⋊C2), C14.32(C4⋊1D4), C14.81(C8⋊C22), (C2×C56).116C22, (C2×C28).453C23, (Q8×C14).82C22, (D4×C14).102C22, (C2×D28).123C22, (C4×Dic7).52C22, (C2×D4⋊D7)⋊21C2, (C2×Q8⋊D7)⋊19C2, C4.10(C2×C7⋊D4), (C2×C14).365(C2×D4), (C2×C7⋊C8).161C22, (C2×C4).542(C22×D7), SmallGroup(448,710)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊9D4
G = < a,b,c | a56=b4=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >
Subgroups: 964 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C14, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C2×SD16, C7⋊C8, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C8⋊3D4, D56, C2×C7⋊C8, C4×Dic7, D14⋊C4, D4⋊D7, Q8⋊D7, C2×C56, C7×SD16, C2×D28, C2×C7⋊D4, D4×C14, Q8×C14, C56⋊C4, C2×D56, C2×D4⋊D7, C28⋊D4, C2×Q8⋊D7, C28.23D4, C14×SD16, C56⋊9D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8⋊C22, C7⋊D4, C22×D7, C8⋊3D4, D4×D7, C2×C7⋊D4, D56⋊C2, C28⋊D4, C56⋊9D4
►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)
(1 110 167 206)(2 67 168 219)(3 80 113 176)(4 93 114 189)(5 106 115 202)(6 63 116 215)(7 76 117 172)(8 89 118 185)(9 102 119 198)(10 59 120 211)(11 72 121 224)(12 85 122 181)(13 98 123 194)(14 111 124 207)(15 68 125 220)(16 81 126 177)(17 94 127 190)(18 107 128 203)(19 64 129 216)(20 77 130 173)(21 90 131 186)(22 103 132 199)(23 60 133 212)(24 73 134 169)(25 86 135 182)(26 99 136 195)(27 112 137 208)(28 69 138 221)(29 82 139 178)(30 95 140 191)(31 108 141 204)(32 65 142 217)(33 78 143 174)(34 91 144 187)(35 104 145 200)(36 61 146 213)(37 74 147 170)(38 87 148 183)(39 100 149 196)(40 57 150 209)(41 70 151 222)(42 83 152 179)(43 96 153 192)(44 109 154 205)(45 66 155 218)(46 79 156 175)(47 92 157 188)(48 105 158 201)(49 62 159 214)(50 75 160 171)(51 88 161 184)(52 101 162 197)(53 58 163 210)(54 71 164 223)(55 84 165 180)(56 97 166 193)
(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(57 203)(58 202)(59 201)(60 200)(61 199)(62 198)(63 197)(64 196)(65 195)(66 194)(67 193)(68 192)(69 191)(70 190)(71 189)(72 188)(73 187)(74 186)(75 185)(76 184)(77 183)(78 182)(79 181)(80 180)(81 179)(82 178)(83 177)(84 176)(85 175)(86 174)(87 173)(88 172)(89 171)(90 170)(91 169)(92 224)(93 223)(94 222)(95 221)(96 220)(97 219)(98 218)(99 217)(100 216)(101 215)(102 214)(103 213)(104 212)(105 211)(106 210)(107 209)(108 208)(109 207)(110 206)(111 205)(112 204)(113 165)(114 164)(115 163)(116 162)(117 161)(118 160)(119 159)(120 158)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)(166 168)
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), (1,110,167,206)(2,67,168,219)(3,80,113,176)(4,93,114,189)(5,106,115,202)(6,63,116,215)(7,76,117,172)(8,89,118,185)(9,102,119,198)(10,59,120,211)(11,72,121,224)(12,85,122,181)(13,98,123,194)(14,111,124,207)(15,68,125,220)(16,81,126,177)(17,94,127,190)(18,107,128,203)(19,64,129,216)(20,77,130,173)(21,90,131,186)(22,103,132,199)(23,60,133,212)(24,73,134,169)(25,86,135,182)(26,99,136,195)(27,112,137,208)(28,69,138,221)(29,82,139,178)(30,95,140,191)(31,108,141,204)(32,65,142,217)(33,78,143,174)(34,91,144,187)(35,104,145,200)(36,61,146,213)(37,74,147,170)(38,87,148,183)(39,100,149,196)(40,57,150,209)(41,70,151,222)(42,83,152,179)(43,96,153,192)(44,109,154,205)(45,66,155,218)(46,79,156,175)(47,92,157,188)(48,105,158,201)(49,62,159,214)(50,75,160,171)(51,88,161,184)(52,101,162,197)(53,58,163,210)(54,71,164,223)(55,84,165,180)(56,97,166,193), (2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(57,203)(58,202)(59,201)(60,200)(61,199)(62,198)(63,197)(64,196)(65,195)(66,194)(67,193)(68,192)(69,191)(70,190)(71,189)(72,188)(73,187)(74,186)(75,185)(76,184)(77,183)(78,182)(79,181)(80,180)(81,179)(82,178)(83,177)(84,176)(85,175)(86,174)(87,173)(88,172)(89,171)(90,170)(91,169)(92,224)(93,223)(94,222)(95,221)(96,220)(97,219)(98,218)(99,217)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)(107,209)(108,208)(109,207)(110,206)(111,205)(112,204)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(166,168)>;
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), (1,110,167,206)(2,67,168,219)(3,80,113,176)(4,93,114,189)(5,106,115,202)(6,63,116,215)(7,76,117,172)(8,89,118,185)(9,102,119,198)(10,59,120,211)(11,72,121,224)(12,85,122,181)(13,98,123,194)(14,111,124,207)(15,68,125,220)(16,81,126,177)(17,94,127,190)(18,107,128,203)(19,64,129,216)(20,77,130,173)(21,90,131,186)(22,103,132,199)(23,60,133,212)(24,73,134,169)(25,86,135,182)(26,99,136,195)(27,112,137,208)(28,69,138,221)(29,82,139,178)(30,95,140,191)(31,108,141,204)(32,65,142,217)(33,78,143,174)(34,91,144,187)(35,104,145,200)(36,61,146,213)(37,74,147,170)(38,87,148,183)(39,100,149,196)(40,57,150,209)(41,70,151,222)(42,83,152,179)(43,96,153,192)(44,109,154,205)(45,66,155,218)(46,79,156,175)(47,92,157,188)(48,105,158,201)(49,62,159,214)(50,75,160,171)(51,88,161,184)(52,101,162,197)(53,58,163,210)(54,71,164,223)(55,84,165,180)(56,97,166,193), (2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(57,203)(58,202)(59,201)(60,200)(61,199)(62,198)(63,197)(64,196)(65,195)(66,194)(67,193)(68,192)(69,191)(70,190)(71,189)(72,188)(73,187)(74,186)(75,185)(76,184)(77,183)(78,182)(79,181)(80,180)(81,179)(82,178)(83,177)(84,176)(85,175)(86,174)(87,173)(88,172)(89,171)(90,170)(91,169)(92,224)(93,223)(94,222)(95,221)(96,220)(97,219)(98,218)(99,217)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)(107,209)(108,208)(109,207)(110,206)(111,205)(112,204)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(166,168) );
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)], [(1,110,167,206),(2,67,168,219),(3,80,113,176),(4,93,114,189),(5,106,115,202),(6,63,116,215),(7,76,117,172),(8,89,118,185),(9,102,119,198),(10,59,120,211),(11,72,121,224),(12,85,122,181),(13,98,123,194),(14,111,124,207),(15,68,125,220),(16,81,126,177),(17,94,127,190),(18,107,128,203),(19,64,129,216),(20,77,130,173),(21,90,131,186),(22,103,132,199),(23,60,133,212),(24,73,134,169),(25,86,135,182),(26,99,136,195),(27,112,137,208),(28,69,138,221),(29,82,139,178),(30,95,140,191),(31,108,141,204),(32,65,142,217),(33,78,143,174),(34,91,144,187),(35,104,145,200),(36,61,146,213),(37,74,147,170),(38,87,148,183),(39,100,149,196),(40,57,150,209),(41,70,151,222),(42,83,152,179),(43,96,153,192),(44,109,154,205),(45,66,155,218),(46,79,156,175),(47,92,157,188),(48,105,158,201),(49,62,159,214),(50,75,160,171),(51,88,161,184),(52,101,162,197),(53,58,163,210),(54,71,164,223),(55,84,165,180),(56,97,166,193)], [(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(57,203),(58,202),(59,201),(60,200),(61,199),(62,198),(63,197),(64,196),(65,195),(66,194),(67,193),(68,192),(69,191),(70,190),(71,189),(72,188),(73,187),(74,186),(75,185),(76,184),(77,183),(78,182),(79,181),(80,180),(81,179),(82,178),(83,177),(84,176),(85,175),(86,174),(87,173),(88,172),(89,171),(90,170),(91,169),(92,224),(93,223),(94,222),(95,221),(96,220),(97,219),(98,218),(99,217),(100,216),(101,215),(102,214),(103,213),(104,212),(105,211),(106,210),(107,209),(108,208),(109,207),(110,206),(111,205),(112,204),(113,165),(114,164),(115,163),(116,162),(117,161),(118,160),(119,159),(120,158),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,140),(166,168)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 56 | 56 | 2 | 2 | 8 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C8⋊C22 | D4×D7 | D4×D7 | D56⋊C2 |
| kernel | C56⋊9D4 | C56⋊C4 | C2×D56 | C2×D4⋊D7 | C28⋊D4 | C2×Q8⋊D7 | C28.23D4 | C14×SD16 | C7⋊C8 | C56 | C2×Dic7 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56⋊9D4 ►in GL6(𝔽113)
| 24 | 112 | 0 | 0 | 0 | 0 |
| 2 | 80 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 76 | 56 | 92 |
| 0 | 0 | 74 | 100 | 38 | 40 |
| 0 | 0 | 108 | 46 | 0 | 0 |
| 0 | 0 | 77 | 100 | 0 | 0 |
| 54 | 85 | 0 | 0 | 0 | 0 |
| 84 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 3 | 7 | 108 |
| 0 | 0 | 55 | 91 | 38 | 50 |
| 0 | 0 | 57 | 17 | 39 | 31 |
| 0 | 0 | 29 | 69 | 63 | 74 |
| 70 | 104 | 0 | 0 | 0 | 0 |
| 17 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 89 | 0 | 0 |
| 0 | 0 | 47 | 15 | 0 | 0 |
| 0 | 0 | 51 | 39 | 112 | 0 |
| 0 | 0 | 20 | 32 | 89 | 1 |
G:=sub<GL(6,GF(113))| [24,2,0,0,0,0,112,80,0,0,0,0,0,0,54,74,108,77,0,0,76,100,46,100,0,0,56,38,0,0,0,0,92,40,0,0],[54,84,0,0,0,0,85,59,0,0,0,0,0,0,22,55,57,29,0,0,3,91,17,69,0,0,7,38,39,63,0,0,108,50,31,74],[70,17,0,0,0,0,104,43,0,0,0,0,0,0,98,47,51,20,0,0,89,15,39,32,0,0,0,0,112,89,0,0,0,0,0,1] >;
C56⋊9D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_9D_4 % in TeX
G:=Group("C56:9D4"); // GroupNames label
G:=SmallGroup(448,710);
// by ID
G=gap.SmallGroup(448,710);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,219,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations