metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊9C4, C8⋊4(C4×D7), C56⋊8(C2×C4), D28⋊6(C2×C4), C4.Q8⋊5D7, C56⋊C4⋊3C2, C7⋊3(D8⋊C4), C14.51(C4×D4), (C2×C8).63D14, D28⋊C4⋊6C2, C4⋊C4.163D14, (C2×D56).13C2, C14.D8⋊18C2, C22.86(D4×D7), C28.34(C4○D4), C2.6(D56⋊C2), C28.45(C22×C4), C4.6(Q8⋊2D7), C14.72(C8⋊C22), (C2×C56).112C22, (C2×C28).285C23, (C2×Dic7).165D4, (C2×D28).77C22, C2.11(D28⋊C4), (C4×Dic7).31C22, C4.42(C2×C4×D7), (C7×C4.Q8)⋊5C2, (C2×C7⋊C8).62C22, (C2×C14).290(C2×D4), (C7×C4⋊C4).78C22, (C2×C4).388(C22×D7), SmallGroup(448,403)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊9C4
G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a43, cbc-1=a42b >
Subgroups: 844 in 132 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, D8⋊C4, D56, C2×C7⋊C8, C4×Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C14.D8, C56⋊C4, C7×C4.Q8, D28⋊C4, C2×D56, D56⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8⋊C22, C4×D7, C22×D7, D8⋊C4, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, D56⋊C2, D56⋊9C4
►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 85)(2 84)(3 83)(4 82)(5 81)(6 80)(7 79)(8 78)(9 77)(10 76)(11 75)(12 74)(13 73)(14 72)(15 71)(16 70)(17 69)(18 68)(19 67)(20 66)(21 65)(22 64)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 101)(42 100)(43 99)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(113 221)(114 220)(115 219)(116 218)(117 217)(118 216)(119 215)(120 214)(121 213)(122 212)(123 211)(124 210)(125 209)(126 208)(127 207)(128 206)(129 205)(130 204)(131 203)(132 202)(133 201)(134 200)(135 199)(136 198)(137 197)(138 196)(139 195)(140 194)(141 193)(142 192)(143 191)(144 190)(145 189)(146 188)(147 187)(148 186)(149 185)(150 184)(151 183)(152 182)(153 181)(154 180)(155 179)(156 178)(157 177)(158 176)(159 175)(160 174)(161 173)(162 172)(163 171)(164 170)(165 169)(166 224)(167 223)(168 222)
(1 115 86 220)(2 158 87 207)(3 145 88 194)(4 132 89 181)(5 119 90 224)(6 162 91 211)(7 149 92 198)(8 136 93 185)(9 123 94 172)(10 166 95 215)(11 153 96 202)(12 140 97 189)(13 127 98 176)(14 114 99 219)(15 157 100 206)(16 144 101 193)(17 131 102 180)(18 118 103 223)(19 161 104 210)(20 148 105 197)(21 135 106 184)(22 122 107 171)(23 165 108 214)(24 152 109 201)(25 139 110 188)(26 126 111 175)(27 113 112 218)(28 156 57 205)(29 143 58 192)(30 130 59 179)(31 117 60 222)(32 160 61 209)(33 147 62 196)(34 134 63 183)(35 121 64 170)(36 164 65 213)(37 151 66 200)(38 138 67 187)(39 125 68 174)(40 168 69 217)(41 155 70 204)(42 142 71 191)(43 129 72 178)(44 116 73 221)(45 159 74 208)(46 146 75 195)(47 133 76 182)(48 120 77 169)(49 163 78 212)(50 150 79 199)(51 137 80 186)(52 124 81 173)(53 167 82 216)(54 154 83 203)(55 141 84 190)(56 128 85 177)
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), (1,85)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,101)(42,100)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(113,221)(114,220)(115,219)(116,218)(117,217)(118,216)(119,215)(120,214)(121,213)(122,212)(123,211)(124,210)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,189)(146,188)(147,187)(148,186)(149,185)(150,184)(151,183)(152,182)(153,181)(154,180)(155,179)(156,178)(157,177)(158,176)(159,175)(160,174)(161,173)(162,172)(163,171)(164,170)(165,169)(166,224)(167,223)(168,222), (1,115,86,220)(2,158,87,207)(3,145,88,194)(4,132,89,181)(5,119,90,224)(6,162,91,211)(7,149,92,198)(8,136,93,185)(9,123,94,172)(10,166,95,215)(11,153,96,202)(12,140,97,189)(13,127,98,176)(14,114,99,219)(15,157,100,206)(16,144,101,193)(17,131,102,180)(18,118,103,223)(19,161,104,210)(20,148,105,197)(21,135,106,184)(22,122,107,171)(23,165,108,214)(24,152,109,201)(25,139,110,188)(26,126,111,175)(27,113,112,218)(28,156,57,205)(29,143,58,192)(30,130,59,179)(31,117,60,222)(32,160,61,209)(33,147,62,196)(34,134,63,183)(35,121,64,170)(36,164,65,213)(37,151,66,200)(38,138,67,187)(39,125,68,174)(40,168,69,217)(41,155,70,204)(42,142,71,191)(43,129,72,178)(44,116,73,221)(45,159,74,208)(46,146,75,195)(47,133,76,182)(48,120,77,169)(49,163,78,212)(50,150,79,199)(51,137,80,186)(52,124,81,173)(53,167,82,216)(54,154,83,203)(55,141,84,190)(56,128,85,177)>;
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), (1,85)(2,84)(3,83)(4,82)(5,81)(6,80)(7,79)(8,78)(9,77)(10,76)(11,75)(12,74)(13,73)(14,72)(15,71)(16,70)(17,69)(18,68)(19,67)(20,66)(21,65)(22,64)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,101)(42,100)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(113,221)(114,220)(115,219)(116,218)(117,217)(118,216)(119,215)(120,214)(121,213)(122,212)(123,211)(124,210)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,189)(146,188)(147,187)(148,186)(149,185)(150,184)(151,183)(152,182)(153,181)(154,180)(155,179)(156,178)(157,177)(158,176)(159,175)(160,174)(161,173)(162,172)(163,171)(164,170)(165,169)(166,224)(167,223)(168,222), (1,115,86,220)(2,158,87,207)(3,145,88,194)(4,132,89,181)(5,119,90,224)(6,162,91,211)(7,149,92,198)(8,136,93,185)(9,123,94,172)(10,166,95,215)(11,153,96,202)(12,140,97,189)(13,127,98,176)(14,114,99,219)(15,157,100,206)(16,144,101,193)(17,131,102,180)(18,118,103,223)(19,161,104,210)(20,148,105,197)(21,135,106,184)(22,122,107,171)(23,165,108,214)(24,152,109,201)(25,139,110,188)(26,126,111,175)(27,113,112,218)(28,156,57,205)(29,143,58,192)(30,130,59,179)(31,117,60,222)(32,160,61,209)(33,147,62,196)(34,134,63,183)(35,121,64,170)(36,164,65,213)(37,151,66,200)(38,138,67,187)(39,125,68,174)(40,168,69,217)(41,155,70,204)(42,142,71,191)(43,129,72,178)(44,116,73,221)(45,159,74,208)(46,146,75,195)(47,133,76,182)(48,120,77,169)(49,163,78,212)(50,150,79,199)(51,137,80,186)(52,124,81,173)(53,167,82,216)(54,154,83,203)(55,141,84,190)(56,128,85,177) );
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)], [(1,85),(2,84),(3,83),(4,82),(5,81),(6,80),(7,79),(8,78),(9,77),(10,76),(11,75),(12,74),(13,73),(14,72),(15,71),(16,70),(17,69),(18,68),(19,67),(20,66),(21,65),(22,64),(23,63),(24,62),(25,61),(26,60),(27,59),(28,58),(29,57),(30,112),(31,111),(32,110),(33,109),(34,108),(35,107),(36,106),(37,105),(38,104),(39,103),(40,102),(41,101),(42,100),(43,99),(44,98),(45,97),(46,96),(47,95),(48,94),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(113,221),(114,220),(115,219),(116,218),(117,217),(118,216),(119,215),(120,214),(121,213),(122,212),(123,211),(124,210),(125,209),(126,208),(127,207),(128,206),(129,205),(130,204),(131,203),(132,202),(133,201),(134,200),(135,199),(136,198),(137,197),(138,196),(139,195),(140,194),(141,193),(142,192),(143,191),(144,190),(145,189),(146,188),(147,187),(148,186),(149,185),(150,184),(151,183),(152,182),(153,181),(154,180),(155,179),(156,178),(157,177),(158,176),(159,175),(160,174),(161,173),(162,172),(163,171),(164,170),(165,169),(166,224),(167,223),(168,222)], [(1,115,86,220),(2,158,87,207),(3,145,88,194),(4,132,89,181),(5,119,90,224),(6,162,91,211),(7,149,92,198),(8,136,93,185),(9,123,94,172),(10,166,95,215),(11,153,96,202),(12,140,97,189),(13,127,98,176),(14,114,99,219),(15,157,100,206),(16,144,101,193),(17,131,102,180),(18,118,103,223),(19,161,104,210),(20,148,105,197),(21,135,106,184),(22,122,107,171),(23,165,108,214),(24,152,109,201),(25,139,110,188),(26,126,111,175),(27,113,112,218),(28,156,57,205),(29,143,58,192),(30,130,59,179),(31,117,60,222),(32,160,61,209),(33,147,62,196),(34,134,63,183),(35,121,64,170),(36,164,65,213),(37,151,66,200),(38,138,67,187),(39,125,68,174),(40,168,69,217),(41,155,70,204),(42,142,71,191),(43,129,72,178),(44,116,73,221),(45,159,74,208),(46,146,75,195),(47,133,76,182),(48,120,77,169),(49,163,78,212),(50,150,79,199),(51,137,80,186),(52,124,81,173),(53,167,82,216),(54,154,83,203),(55,141,84,190),(56,128,85,177)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | C8⋊C22 | Q8⋊2D7 | D4×D7 | D56⋊C2 |
| kernel | D56⋊9C4 | C14.D8 | C56⋊C4 | C7×C4.Q8 | D28⋊C4 | C2×D56 | D56 | C2×Dic7 | C4.Q8 | C28 | C4⋊C4 | C2×C8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 2 | 6 | 3 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of D56⋊9C4 ►in GL6(𝔽113)
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 7 | 0 | 50 |
| 0 | 0 | 99 | 36 | 85 | 41 |
| 0 | 0 | 44 | 96 | 0 | 0 |
| 0 | 0 | 78 | 73 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 65 | 48 | 0 |
| 0 | 0 | 21 | 85 | 71 | 65 |
| 0 | 0 | 40 | 0 | 43 | 65 |
| 0 | 0 | 78 | 73 | 22 | 70 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 12 | 50 | 12 |
| 0 | 0 | 89 | 104 | 106 | 96 |
| 0 | 0 | 34 | 87 | 55 | 101 |
| 0 | 0 | 86 | 85 | 70 | 110 |
G:=sub<GL(6,GF(113))| [0,1,0,0,0,0,112,0,0,0,0,0,0,0,35,99,44,78,0,0,7,36,96,73,0,0,0,85,0,0,0,0,50,41,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,21,40,78,0,0,65,85,0,73,0,0,48,71,43,22,0,0,0,65,65,70],[98,0,0,0,0,0,0,15,0,0,0,0,0,0,70,89,34,86,0,0,12,104,87,85,0,0,50,106,55,70,0,0,12,96,101,110] >;
D56⋊9C4 in GAP, Magma, Sage, TeX
D_{56}\rtimes_9C_4 % in TeX
G:=Group("D56:9C4"); // GroupNames label
G:=SmallGroup(448,403);
// by ID
G=gap.SmallGroup(448,403);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,219,58,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^42*b>;
// generators/relations