metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.43D4, C7⋊C8.20D4, C4.24(D4×D7), (C2×SD16)⋊9D7, (C14×SD16)⋊7C2, (C8×Dic7)⋊10C2, (C2×D4).71D14, C28.175(C2×D4), (C2×C8).262D14, C7⋊4(C8.12D4), C8.20(C7⋊D4), (C2×Q8).53D14, C14.62(C4○D8), C28.17D4⋊6C2, C28.23D4⋊4C2, C22.265(D4×D7), C2.21(C28⋊D4), C14.30(C4⋊1D4), (C2×C28).445C23, (C2×C56).163C22, (C2×Dic7).113D4, (D4×C14).94C22, (Q8×C14).75C22, (C2×D28).119C22, C2.28(SD16⋊3D7), (C4×Dic7).241C22, (C2×Dic14).126C22, C4.8(C2×C7⋊D4), (C2×D4⋊D7).9C2, (C2×C56⋊C2)⋊29C2, (C2×C7⋊Q16)⋊18C2, (C2×C14).357(C2×D4), (C2×C7⋊C8).273C22, (C2×C4).534(C22×D7), SmallGroup(448,702)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.43D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a41, cac-1=a27, cbc-1=a28b-1 >
Subgroups: 708 in 130 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, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×SD16, C2×Q16, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C8.12D4, C56⋊C2, C2×C7⋊C8, C4×Dic7, D14⋊C4, D4⋊D7, C7⋊Q16, C23.D7, C2×C56, C7×SD16, C2×Dic14, C2×D28, D4×C14, Q8×C14, C8×Dic7, C2×C56⋊C2, C2×D4⋊D7, C28.17D4, C2×C7⋊Q16, C28.23D4, C14×SD16, C56.43D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C4○D8, C7⋊D4, C22×D7, C8.12D4, D4×D7, C2×C7⋊D4, SD16⋊3D7, C28⋊D4, C56.43D4
►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)
(1 146 63 204)(2 131 64 189)(3 116 65 174)(4 157 66 215)(5 142 67 200)(6 127 68 185)(7 168 69 170)(8 153 70 211)(9 138 71 196)(10 123 72 181)(11 164 73 222)(12 149 74 207)(13 134 75 192)(14 119 76 177)(15 160 77 218)(16 145 78 203)(17 130 79 188)(18 115 80 173)(19 156 81 214)(20 141 82 199)(21 126 83 184)(22 167 84 169)(23 152 85 210)(24 137 86 195)(25 122 87 180)(26 163 88 221)(27 148 89 206)(28 133 90 191)(29 118 91 176)(30 159 92 217)(31 144 93 202)(32 129 94 187)(33 114 95 172)(34 155 96 213)(35 140 97 198)(36 125 98 183)(37 166 99 224)(38 151 100 209)(39 136 101 194)(40 121 102 179)(41 162 103 220)(42 147 104 205)(43 132 105 190)(44 117 106 175)(45 158 107 216)(46 143 108 201)(47 128 109 186)(48 113 110 171)(49 154 111 212)(50 139 112 197)(51 124 57 182)(52 165 58 223)(53 150 59 208)(54 135 60 193)(55 120 61 178)(56 161 62 219)
(1 8 29 36)(2 35 30 7)(3 6 31 34)(4 33 32 5)(9 56 37 28)(10 27 38 55)(11 54 39 26)(12 25 40 53)(13 52 41 24)(14 23 42 51)(15 50 43 22)(16 21 44 49)(17 48 45 20)(18 19 46 47)(57 76 85 104)(58 103 86 75)(59 74 87 102)(60 101 88 73)(61 72 89 100)(62 99 90 71)(63 70 91 98)(64 97 92 69)(65 68 93 96)(66 95 94 67)(77 112 105 84)(78 83 106 111)(79 110 107 82)(80 81 108 109)(113 188 141 216)(114 215 142 187)(115 186 143 214)(116 213 144 185)(117 184 145 212)(118 211 146 183)(119 182 147 210)(120 209 148 181)(121 180 149 208)(122 207 150 179)(123 178 151 206)(124 205 152 177)(125 176 153 204)(126 203 154 175)(127 174 155 202)(128 201 156 173)(129 172 157 200)(130 199 158 171)(131 170 159 198)(132 197 160 169)(133 224 161 196)(134 195 162 223)(135 222 163 194)(136 193 164 221)(137 220 165 192)(138 191 166 219)(139 218 167 190)(140 189 168 217)
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), (1,146,63,204)(2,131,64,189)(3,116,65,174)(4,157,66,215)(5,142,67,200)(6,127,68,185)(7,168,69,170)(8,153,70,211)(9,138,71,196)(10,123,72,181)(11,164,73,222)(12,149,74,207)(13,134,75,192)(14,119,76,177)(15,160,77,218)(16,145,78,203)(17,130,79,188)(18,115,80,173)(19,156,81,214)(20,141,82,199)(21,126,83,184)(22,167,84,169)(23,152,85,210)(24,137,86,195)(25,122,87,180)(26,163,88,221)(27,148,89,206)(28,133,90,191)(29,118,91,176)(30,159,92,217)(31,144,93,202)(32,129,94,187)(33,114,95,172)(34,155,96,213)(35,140,97,198)(36,125,98,183)(37,166,99,224)(38,151,100,209)(39,136,101,194)(40,121,102,179)(41,162,103,220)(42,147,104,205)(43,132,105,190)(44,117,106,175)(45,158,107,216)(46,143,108,201)(47,128,109,186)(48,113,110,171)(49,154,111,212)(50,139,112,197)(51,124,57,182)(52,165,58,223)(53,150,59,208)(54,135,60,193)(55,120,61,178)(56,161,62,219), (1,8,29,36)(2,35,30,7)(3,6,31,34)(4,33,32,5)(9,56,37,28)(10,27,38,55)(11,54,39,26)(12,25,40,53)(13,52,41,24)(14,23,42,51)(15,50,43,22)(16,21,44,49)(17,48,45,20)(18,19,46,47)(57,76,85,104)(58,103,86,75)(59,74,87,102)(60,101,88,73)(61,72,89,100)(62,99,90,71)(63,70,91,98)(64,97,92,69)(65,68,93,96)(66,95,94,67)(77,112,105,84)(78,83,106,111)(79,110,107,82)(80,81,108,109)(113,188,141,216)(114,215,142,187)(115,186,143,214)(116,213,144,185)(117,184,145,212)(118,211,146,183)(119,182,147,210)(120,209,148,181)(121,180,149,208)(122,207,150,179)(123,178,151,206)(124,205,152,177)(125,176,153,204)(126,203,154,175)(127,174,155,202)(128,201,156,173)(129,172,157,200)(130,199,158,171)(131,170,159,198)(132,197,160,169)(133,224,161,196)(134,195,162,223)(135,222,163,194)(136,193,164,221)(137,220,165,192)(138,191,166,219)(139,218,167,190)(140,189,168,217)>;
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), (1,146,63,204)(2,131,64,189)(3,116,65,174)(4,157,66,215)(5,142,67,200)(6,127,68,185)(7,168,69,170)(8,153,70,211)(9,138,71,196)(10,123,72,181)(11,164,73,222)(12,149,74,207)(13,134,75,192)(14,119,76,177)(15,160,77,218)(16,145,78,203)(17,130,79,188)(18,115,80,173)(19,156,81,214)(20,141,82,199)(21,126,83,184)(22,167,84,169)(23,152,85,210)(24,137,86,195)(25,122,87,180)(26,163,88,221)(27,148,89,206)(28,133,90,191)(29,118,91,176)(30,159,92,217)(31,144,93,202)(32,129,94,187)(33,114,95,172)(34,155,96,213)(35,140,97,198)(36,125,98,183)(37,166,99,224)(38,151,100,209)(39,136,101,194)(40,121,102,179)(41,162,103,220)(42,147,104,205)(43,132,105,190)(44,117,106,175)(45,158,107,216)(46,143,108,201)(47,128,109,186)(48,113,110,171)(49,154,111,212)(50,139,112,197)(51,124,57,182)(52,165,58,223)(53,150,59,208)(54,135,60,193)(55,120,61,178)(56,161,62,219), (1,8,29,36)(2,35,30,7)(3,6,31,34)(4,33,32,5)(9,56,37,28)(10,27,38,55)(11,54,39,26)(12,25,40,53)(13,52,41,24)(14,23,42,51)(15,50,43,22)(16,21,44,49)(17,48,45,20)(18,19,46,47)(57,76,85,104)(58,103,86,75)(59,74,87,102)(60,101,88,73)(61,72,89,100)(62,99,90,71)(63,70,91,98)(64,97,92,69)(65,68,93,96)(66,95,94,67)(77,112,105,84)(78,83,106,111)(79,110,107,82)(80,81,108,109)(113,188,141,216)(114,215,142,187)(115,186,143,214)(116,213,144,185)(117,184,145,212)(118,211,146,183)(119,182,147,210)(120,209,148,181)(121,180,149,208)(122,207,150,179)(123,178,151,206)(124,205,152,177)(125,176,153,204)(126,203,154,175)(127,174,155,202)(128,201,156,173)(129,172,157,200)(130,199,158,171)(131,170,159,198)(132,197,160,169)(133,224,161,196)(134,195,162,223)(135,222,163,194)(136,193,164,221)(137,220,165,192)(138,191,166,219)(139,218,167,190)(140,189,168,217) );
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)], [(1,146,63,204),(2,131,64,189),(3,116,65,174),(4,157,66,215),(5,142,67,200),(6,127,68,185),(7,168,69,170),(8,153,70,211),(9,138,71,196),(10,123,72,181),(11,164,73,222),(12,149,74,207),(13,134,75,192),(14,119,76,177),(15,160,77,218),(16,145,78,203),(17,130,79,188),(18,115,80,173),(19,156,81,214),(20,141,82,199),(21,126,83,184),(22,167,84,169),(23,152,85,210),(24,137,86,195),(25,122,87,180),(26,163,88,221),(27,148,89,206),(28,133,90,191),(29,118,91,176),(30,159,92,217),(31,144,93,202),(32,129,94,187),(33,114,95,172),(34,155,96,213),(35,140,97,198),(36,125,98,183),(37,166,99,224),(38,151,100,209),(39,136,101,194),(40,121,102,179),(41,162,103,220),(42,147,104,205),(43,132,105,190),(44,117,106,175),(45,158,107,216),(46,143,108,201),(47,128,109,186),(48,113,110,171),(49,154,111,212),(50,139,112,197),(51,124,57,182),(52,165,58,223),(53,150,59,208),(54,135,60,193),(55,120,61,178),(56,161,62,219)], [(1,8,29,36),(2,35,30,7),(3,6,31,34),(4,33,32,5),(9,56,37,28),(10,27,38,55),(11,54,39,26),(12,25,40,53),(13,52,41,24),(14,23,42,51),(15,50,43,22),(16,21,44,49),(17,48,45,20),(18,19,46,47),(57,76,85,104),(58,103,86,75),(59,74,87,102),(60,101,88,73),(61,72,89,100),(62,99,90,71),(63,70,91,98),(64,97,92,69),(65,68,93,96),(66,95,94,67),(77,112,105,84),(78,83,106,111),(79,110,107,82),(80,81,108,109),(113,188,141,216),(114,215,142,187),(115,186,143,214),(116,213,144,185),(117,184,145,212),(118,211,146,183),(119,182,147,210),(120,209,148,181),(121,180,149,208),(122,207,150,179),(123,178,151,206),(124,205,152,177),(125,176,153,204),(126,203,154,175),(127,174,155,202),(128,201,156,173),(129,172,157,200),(130,199,158,171),(131,170,159,198),(132,197,160,169),(133,224,161,196),(134,195,162,223),(135,222,163,194),(136,193,164,221),(137,220,165,192),(138,191,166,219),(139,218,167,190),(140,189,168,217)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 56 | 2 | 2 | 8 | 14 | 14 | 14 | 14 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | D4×D7 | D4×D7 | SD16⋊3D7 |
| kernel | C56.43D4 | C8×Dic7 | C2×C56⋊C2 | C2×D4⋊D7 | C28.17D4 | C2×C7⋊Q16 | C28.23D4 | C14×SD16 | C7⋊C8 | C56 | C2×Dic7 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C14 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 8 | 12 | 3 | 3 | 12 |
Matrix representation of C56.43D4 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 100 | 0 |
| 79 | 108 | 0 | 0 |
| 28 | 34 | 0 | 0 |
| 0 | 0 | 15 | 30 |
| 0 | 0 | 98 | 98 |
| 112 | 0 | 0 | 0 |
| 104 | 1 | 0 | 0 |
| 0 | 0 | 87 | 87 |
| 0 | 0 | 13 | 26 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,26,100,0,0,26,0],[79,28,0,0,108,34,0,0,0,0,15,98,0,0,30,98],[112,104,0,0,0,1,0,0,0,0,87,13,0,0,87,26] >;
C56.43D4 in GAP, Magma, Sage, TeX
C_{56}._{43}D_4 % in TeX
G:=Group("C56.43D4"); // GroupNames label
G:=SmallGroup(448,702);
// by ID
G=gap.SmallGroup(448,702);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,1094,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^41,c*a*c^-1=a^27,c*b*c^-1=a^28*b^-1>;
// generators/relations