direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C56, (C4×C8)⋊4C14, (C4×C56)⋊9C2, C4⋊1(C2×C56), C28⋊6(C2×C8), C4⋊C8⋊18C14, C2.3(D4×C28), C4⋊C4.11C28, (C22×C8)⋊5C14, (C22×C56)⋊9C2, C22⋊1(C2×C56), C4.79(D4×C14), C22⋊C8⋊15C14, (D4×C14).23C4, (C4×D4).14C14, (D4×C28).29C2, (C2×D4).11C28, C14.111(C4×D4), C28.484(C2×D4), C22⋊C4.7C28, C2.4(C22×C56), C14.48(C8○D4), C23.18(C2×C28), C42.68(C2×C14), C14.33(C22×C8), C28.353(C4○D4), (C4×C28).353C22, (C2×C56).361C22, (C2×C28).990C23, C22.22(C22×C28), (C22×C28).499C22, (C7×C4⋊C8)⋊37C2, (C2×C14)⋊4(C2×C8), C2.2(C7×C8○D4), (C7×C4⋊C4).23C4, C4.51(C7×C4○D4), (C7×C22⋊C8)⋊32C2, (C2×C4).36(C2×C28), (C2×C8).107(C2×C14), (C2×C28).212(C2×C4), (C7×C22⋊C4).14C4, (C22×C14).85(C2×C4), (C22×C4).95(C2×C14), (C2×C14).240(C22×C4), (C2×C4).158(C22×C14), SmallGroup(448,842)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C56
G = < a,b,c | a56=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 178 in 134 conjugacy classes, 90 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, C22×C4, C2×D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C56, C56, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C8×D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C2×C56, C22×C28, D4×C14, C4×C56, C7×C22⋊C8, C7×C4⋊C8, D4×C28, C22×C56, D4×C56
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C23, C14, C2×C8, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C22×C8, C8○D4, C56, C2×C28, C7×D4, C22×C14, C8×D4, C2×C56, C22×C28, D4×C14, C7×C4○D4, D4×C28, C22×C56, C7×C8○D4, D4×C56
►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)
(1 158 85 209)(2 159 86 210)(3 160 87 211)(4 161 88 212)(5 162 89 213)(6 163 90 214)(7 164 91 215)(8 165 92 216)(9 166 93 217)(10 167 94 218)(11 168 95 219)(12 113 96 220)(13 114 97 221)(14 115 98 222)(15 116 99 223)(16 117 100 224)(17 118 101 169)(18 119 102 170)(19 120 103 171)(20 121 104 172)(21 122 105 173)(22 123 106 174)(23 124 107 175)(24 125 108 176)(25 126 109 177)(26 127 110 178)(27 128 111 179)(28 129 112 180)(29 130 57 181)(30 131 58 182)(31 132 59 183)(32 133 60 184)(33 134 61 185)(34 135 62 186)(35 136 63 187)(36 137 64 188)(37 138 65 189)(38 139 66 190)(39 140 67 191)(40 141 68 192)(41 142 69 193)(42 143 70 194)(43 144 71 195)(44 145 72 196)(45 146 73 197)(46 147 74 198)(47 148 75 199)(48 149 76 200)(49 150 77 201)(50 151 78 202)(51 152 79 203)(52 153 80 204)(53 154 81 205)(54 155 82 206)(55 156 83 207)(56 157 84 208)
(113 220)(114 221)(115 222)(116 223)(117 224)(118 169)(119 170)(120 171)(121 172)(122 173)(123 174)(124 175)(125 176)(126 177)(127 178)(128 179)(129 180)(130 181)(131 182)(132 183)(133 184)(134 185)(135 186)(136 187)(137 188)(138 189)(139 190)(140 191)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)
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), (1,158,85,209)(2,159,86,210)(3,160,87,211)(4,161,88,212)(5,162,89,213)(6,163,90,214)(7,164,91,215)(8,165,92,216)(9,166,93,217)(10,167,94,218)(11,168,95,219)(12,113,96,220)(13,114,97,221)(14,115,98,222)(15,116,99,223)(16,117,100,224)(17,118,101,169)(18,119,102,170)(19,120,103,171)(20,121,104,172)(21,122,105,173)(22,123,106,174)(23,124,107,175)(24,125,108,176)(25,126,109,177)(26,127,110,178)(27,128,111,179)(28,129,112,180)(29,130,57,181)(30,131,58,182)(31,132,59,183)(32,133,60,184)(33,134,61,185)(34,135,62,186)(35,136,63,187)(36,137,64,188)(37,138,65,189)(38,139,66,190)(39,140,67,191)(40,141,68,192)(41,142,69,193)(42,143,70,194)(43,144,71,195)(44,145,72,196)(45,146,73,197)(46,147,74,198)(47,148,75,199)(48,149,76,200)(49,150,77,201)(50,151,78,202)(51,152,79,203)(52,153,80,204)(53,154,81,205)(54,155,82,206)(55,156,83,207)(56,157,84,208), (113,220)(114,221)(115,222)(116,223)(117,224)(118,169)(119,170)(120,171)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)>;
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), (1,158,85,209)(2,159,86,210)(3,160,87,211)(4,161,88,212)(5,162,89,213)(6,163,90,214)(7,164,91,215)(8,165,92,216)(9,166,93,217)(10,167,94,218)(11,168,95,219)(12,113,96,220)(13,114,97,221)(14,115,98,222)(15,116,99,223)(16,117,100,224)(17,118,101,169)(18,119,102,170)(19,120,103,171)(20,121,104,172)(21,122,105,173)(22,123,106,174)(23,124,107,175)(24,125,108,176)(25,126,109,177)(26,127,110,178)(27,128,111,179)(28,129,112,180)(29,130,57,181)(30,131,58,182)(31,132,59,183)(32,133,60,184)(33,134,61,185)(34,135,62,186)(35,136,63,187)(36,137,64,188)(37,138,65,189)(38,139,66,190)(39,140,67,191)(40,141,68,192)(41,142,69,193)(42,143,70,194)(43,144,71,195)(44,145,72,196)(45,146,73,197)(46,147,74,198)(47,148,75,199)(48,149,76,200)(49,150,77,201)(50,151,78,202)(51,152,79,203)(52,153,80,204)(53,154,81,205)(54,155,82,206)(55,156,83,207)(56,157,84,208), (113,220)(114,221)(115,222)(116,223)(117,224)(118,169)(119,170)(120,171)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219) );
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)], [(1,158,85,209),(2,159,86,210),(3,160,87,211),(4,161,88,212),(5,162,89,213),(6,163,90,214),(7,164,91,215),(8,165,92,216),(9,166,93,217),(10,167,94,218),(11,168,95,219),(12,113,96,220),(13,114,97,221),(14,115,98,222),(15,116,99,223),(16,117,100,224),(17,118,101,169),(18,119,102,170),(19,120,103,171),(20,121,104,172),(21,122,105,173),(22,123,106,174),(23,124,107,175),(24,125,108,176),(25,126,109,177),(26,127,110,178),(27,128,111,179),(28,129,112,180),(29,130,57,181),(30,131,58,182),(31,132,59,183),(32,133,60,184),(33,134,61,185),(34,135,62,186),(35,136,63,187),(36,137,64,188),(37,138,65,189),(38,139,66,190),(39,140,67,191),(40,141,68,192),(41,142,69,193),(42,143,70,194),(43,144,71,195),(44,145,72,196),(45,146,73,197),(46,147,74,198),(47,148,75,199),(48,149,76,200),(49,150,77,201),(50,151,78,202),(51,152,79,203),(52,153,80,204),(53,154,81,205),(54,155,82,206),(55,156,83,207),(56,157,84,208)], [(113,220),(114,221),(115,222),(116,223),(117,224),(118,169),(119,170),(120,171),(121,172),(122,173),(123,174),(124,175),(125,176),(126,177),(127,178),(128,179),(129,180),(130,181),(131,182),(132,183),(133,184),(134,185),(135,186),(136,187),(137,188),(138,189),(139,190),(140,191),(141,192),(142,193),(143,194),(144,195),(145,196),(146,197),(147,198),(148,199),(149,200),(150,201),(151,202),(152,203),(153,204),(154,205),(155,206),(156,207),(157,208),(158,209),(159,210),(160,211),(161,212),(162,213),(163,214),(164,215),(165,216),(166,217),(167,218),(168,219)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28BT | 56A | ··· | 56AV | 56AW | ··· | 56DP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | C56 | D4 | C4○D4 | C8○D4 | C7×D4 | C7×C4○D4 | C7×C8○D4 |
| kernel | D4×C56 | C4×C56 | C7×C22⋊C8 | C7×C4⋊C8 | D4×C28 | C22×C56 | C7×C22⋊C4 | C7×C4⋊C4 | D4×C14 | C8×D4 | C7×D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C56 | C28 | C14 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 6 | 16 | 6 | 12 | 6 | 6 | 12 | 24 | 12 | 12 | 96 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of D4×C56 ►in GL3(𝔽113) generated by
| 18 | 0 | 0 |
| 0 | 97 | 0 |
| 0 | 0 | 97 |
| 1 | 0 | 0 |
| 0 | 112 | 111 |
| 0 | 1 | 1 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 112 | 112 |
G:=sub<GL(3,GF(113))| [18,0,0,0,97,0,0,0,97],[1,0,0,0,112,1,0,111,1],[112,0,0,0,1,112,0,0,112] >;
D4×C56 in GAP, Magma, Sage, TeX
D_4\times C_{56} % in TeX
G:=Group("D4xC56"); // GroupNames label
G:=SmallGroup(448,842);
// by ID
G=gap.SmallGroup(448,842);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,604,124]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations