metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.82D4, C22⋊1Dic28, C23.26D28, (C2×C14)⋊5Q16, C56⋊1C4⋊15C2, (C2×C4).69D28, (C22×C8).9D7, (C2×Dic28)⋊9C2, (C2×C8).309D14, (C2×C28).357D4, C28.414(C2×D4), C8.39(C7⋊D4), C7⋊4(C8.18D4), C14.11(C2×Q16), C14.19(C4○D8), C28.44D4⋊3C2, (C22×C56).15C2, C2.11(C2×Dic28), C28.229(C4○D4), C4.113(C4○D28), C2.20(C28⋊7D4), C14.72(C4⋊D4), (C2×C28).770C23, (C2×C56).381C22, C28.48D4.5C2, (C22×C4).432D14, C22.133(C2×D28), (C22×C14).142D4, C4⋊Dic7.25C22, C2.19(D56⋊7C2), (C22×C28).520C22, (C2×Dic14).18C22, C4.107(C2×C7⋊D4), (C2×C14).160(C2×D4), (C2×C4).718(C22×D7), SmallGroup(448,650)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.82D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=cac-1=a-1, cbc-1=a28b-1 >
Subgroups: 516 in 114 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C56, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C8.18D4, Dic28, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C2×C56, C2×Dic14, C22×C28, C28.44D4, C56⋊1C4, C2×Dic28, C28.48D4, C22×C56, C56.82D4
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C4⋊D4, C2×Q16, C4○D8, D28, C7⋊D4, C22×D7, C8.18D4, Dic28, C2×D28, C4○D28, C2×C7⋊D4, D56⋊7C2, C2×Dic28, C28⋊7D4, C56.82D4
►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)
(1 205 152 85)(2 204 153 84)(3 203 154 83)(4 202 155 82)(5 201 156 81)(6 200 157 80)(7 199 158 79)(8 198 159 78)(9 197 160 77)(10 196 161 76)(11 195 162 75)(12 194 163 74)(13 193 164 73)(14 192 165 72)(15 191 166 71)(16 190 167 70)(17 189 168 69)(18 188 113 68)(19 187 114 67)(20 186 115 66)(21 185 116 65)(22 184 117 64)(23 183 118 63)(24 182 119 62)(25 181 120 61)(26 180 121 60)(27 179 122 59)(28 178 123 58)(29 177 124 57)(30 176 125 112)(31 175 126 111)(32 174 127 110)(33 173 128 109)(34 172 129 108)(35 171 130 107)(36 170 131 106)(37 169 132 105)(38 224 133 104)(39 223 134 103)(40 222 135 102)(41 221 136 101)(42 220 137 100)(43 219 138 99)(44 218 139 98)(45 217 140 97)(46 216 141 96)(47 215 142 95)(48 214 143 94)(49 213 144 93)(50 212 145 92)(51 211 146 91)(52 210 147 90)(53 209 148 89)(54 208 149 88)(55 207 150 87)(56 206 151 86)
(1 57 29 85)(2 112 30 84)(3 111 31 83)(4 110 32 82)(5 109 33 81)(6 108 34 80)(7 107 35 79)(8 106 36 78)(9 105 37 77)(10 104 38 76)(11 103 39 75)(12 102 40 74)(13 101 41 73)(14 100 42 72)(15 99 43 71)(16 98 44 70)(17 97 45 69)(18 96 46 68)(19 95 47 67)(20 94 48 66)(21 93 49 65)(22 92 50 64)(23 91 51 63)(24 90 52 62)(25 89 53 61)(26 88 54 60)(27 87 55 59)(28 86 56 58)(113 216 141 188)(114 215 142 187)(115 214 143 186)(116 213 144 185)(117 212 145 184)(118 211 146 183)(119 210 147 182)(120 209 148 181)(121 208 149 180)(122 207 150 179)(123 206 151 178)(124 205 152 177)(125 204 153 176)(126 203 154 175)(127 202 155 174)(128 201 156 173)(129 200 157 172)(130 199 158 171)(131 198 159 170)(132 197 160 169)(133 196 161 224)(134 195 162 223)(135 194 163 222)(136 193 164 221)(137 192 165 220)(138 191 166 219)(139 190 167 218)(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), (1,205,152,85)(2,204,153,84)(3,203,154,83)(4,202,155,82)(5,201,156,81)(6,200,157,80)(7,199,158,79)(8,198,159,78)(9,197,160,77)(10,196,161,76)(11,195,162,75)(12,194,163,74)(13,193,164,73)(14,192,165,72)(15,191,166,71)(16,190,167,70)(17,189,168,69)(18,188,113,68)(19,187,114,67)(20,186,115,66)(21,185,116,65)(22,184,117,64)(23,183,118,63)(24,182,119,62)(25,181,120,61)(26,180,121,60)(27,179,122,59)(28,178,123,58)(29,177,124,57)(30,176,125,112)(31,175,126,111)(32,174,127,110)(33,173,128,109)(34,172,129,108)(35,171,130,107)(36,170,131,106)(37,169,132,105)(38,224,133,104)(39,223,134,103)(40,222,135,102)(41,221,136,101)(42,220,137,100)(43,219,138,99)(44,218,139,98)(45,217,140,97)(46,216,141,96)(47,215,142,95)(48,214,143,94)(49,213,144,93)(50,212,145,92)(51,211,146,91)(52,210,147,90)(53,209,148,89)(54,208,149,88)(55,207,150,87)(56,206,151,86), (1,57,29,85)(2,112,30,84)(3,111,31,83)(4,110,32,82)(5,109,33,81)(6,108,34,80)(7,107,35,79)(8,106,36,78)(9,105,37,77)(10,104,38,76)(11,103,39,75)(12,102,40,74)(13,101,41,73)(14,100,42,72)(15,99,43,71)(16,98,44,70)(17,97,45,69)(18,96,46,68)(19,95,47,67)(20,94,48,66)(21,93,49,65)(22,92,50,64)(23,91,51,63)(24,90,52,62)(25,89,53,61)(26,88,54,60)(27,87,55,59)(28,86,56,58)(113,216,141,188)(114,215,142,187)(115,214,143,186)(116,213,144,185)(117,212,145,184)(118,211,146,183)(119,210,147,182)(120,209,148,181)(121,208,149,180)(122,207,150,179)(123,206,151,178)(124,205,152,177)(125,204,153,176)(126,203,154,175)(127,202,155,174)(128,201,156,173)(129,200,157,172)(130,199,158,171)(131,198,159,170)(132,197,160,169)(133,196,161,224)(134,195,162,223)(135,194,163,222)(136,193,164,221)(137,192,165,220)(138,191,166,219)(139,190,167,218)(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), (1,205,152,85)(2,204,153,84)(3,203,154,83)(4,202,155,82)(5,201,156,81)(6,200,157,80)(7,199,158,79)(8,198,159,78)(9,197,160,77)(10,196,161,76)(11,195,162,75)(12,194,163,74)(13,193,164,73)(14,192,165,72)(15,191,166,71)(16,190,167,70)(17,189,168,69)(18,188,113,68)(19,187,114,67)(20,186,115,66)(21,185,116,65)(22,184,117,64)(23,183,118,63)(24,182,119,62)(25,181,120,61)(26,180,121,60)(27,179,122,59)(28,178,123,58)(29,177,124,57)(30,176,125,112)(31,175,126,111)(32,174,127,110)(33,173,128,109)(34,172,129,108)(35,171,130,107)(36,170,131,106)(37,169,132,105)(38,224,133,104)(39,223,134,103)(40,222,135,102)(41,221,136,101)(42,220,137,100)(43,219,138,99)(44,218,139,98)(45,217,140,97)(46,216,141,96)(47,215,142,95)(48,214,143,94)(49,213,144,93)(50,212,145,92)(51,211,146,91)(52,210,147,90)(53,209,148,89)(54,208,149,88)(55,207,150,87)(56,206,151,86), (1,57,29,85)(2,112,30,84)(3,111,31,83)(4,110,32,82)(5,109,33,81)(6,108,34,80)(7,107,35,79)(8,106,36,78)(9,105,37,77)(10,104,38,76)(11,103,39,75)(12,102,40,74)(13,101,41,73)(14,100,42,72)(15,99,43,71)(16,98,44,70)(17,97,45,69)(18,96,46,68)(19,95,47,67)(20,94,48,66)(21,93,49,65)(22,92,50,64)(23,91,51,63)(24,90,52,62)(25,89,53,61)(26,88,54,60)(27,87,55,59)(28,86,56,58)(113,216,141,188)(114,215,142,187)(115,214,143,186)(116,213,144,185)(117,212,145,184)(118,211,146,183)(119,210,147,182)(120,209,148,181)(121,208,149,180)(122,207,150,179)(123,206,151,178)(124,205,152,177)(125,204,153,176)(126,203,154,175)(127,202,155,174)(128,201,156,173)(129,200,157,172)(130,199,158,171)(131,198,159,170)(132,197,160,169)(133,196,161,224)(134,195,162,223)(135,194,163,222)(136,193,164,221)(137,192,165,220)(138,191,166,219)(139,190,167,218)(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)], [(1,205,152,85),(2,204,153,84),(3,203,154,83),(4,202,155,82),(5,201,156,81),(6,200,157,80),(7,199,158,79),(8,198,159,78),(9,197,160,77),(10,196,161,76),(11,195,162,75),(12,194,163,74),(13,193,164,73),(14,192,165,72),(15,191,166,71),(16,190,167,70),(17,189,168,69),(18,188,113,68),(19,187,114,67),(20,186,115,66),(21,185,116,65),(22,184,117,64),(23,183,118,63),(24,182,119,62),(25,181,120,61),(26,180,121,60),(27,179,122,59),(28,178,123,58),(29,177,124,57),(30,176,125,112),(31,175,126,111),(32,174,127,110),(33,173,128,109),(34,172,129,108),(35,171,130,107),(36,170,131,106),(37,169,132,105),(38,224,133,104),(39,223,134,103),(40,222,135,102),(41,221,136,101),(42,220,137,100),(43,219,138,99),(44,218,139,98),(45,217,140,97),(46,216,141,96),(47,215,142,95),(48,214,143,94),(49,213,144,93),(50,212,145,92),(51,211,146,91),(52,210,147,90),(53,209,148,89),(54,208,149,88),(55,207,150,87),(56,206,151,86)], [(1,57,29,85),(2,112,30,84),(3,111,31,83),(4,110,32,82),(5,109,33,81),(6,108,34,80),(7,107,35,79),(8,106,36,78),(9,105,37,77),(10,104,38,76),(11,103,39,75),(12,102,40,74),(13,101,41,73),(14,100,42,72),(15,99,43,71),(16,98,44,70),(17,97,45,69),(18,96,46,68),(19,95,47,67),(20,94,48,66),(21,93,49,65),(22,92,50,64),(23,91,51,63),(24,90,52,62),(25,89,53,61),(26,88,54,60),(27,87,55,59),(28,86,56,58),(113,216,141,188),(114,215,142,187),(115,214,143,186),(116,213,144,185),(117,212,145,184),(118,211,146,183),(119,210,147,182),(120,209,148,181),(121,208,149,180),(122,207,150,179),(123,206,151,178),(124,205,152,177),(125,204,153,176),(126,203,154,175),(127,202,155,174),(128,201,156,173),(129,200,157,172),(130,199,158,171),(131,198,159,170),(132,197,160,169),(133,196,161,224),(134,195,162,223),(135,194,163,222),(136,193,164,221),(137,192,165,220),(138,191,166,219),(139,190,167,218),(140,189,168,217)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | Q16 | D14 | D14 | C4○D8 | C7⋊D4 | D28 | D28 | C4○D28 | Dic28 | D56⋊7C2 |
| kernel | C56.82D4 | C28.44D4 | C56⋊1C4 | C2×Dic28 | C28.48D4 | C22×C56 | C56 | C2×C28 | C22×C14 | C22×C8 | C28 | C2×C14 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C56.82D4 ►in GL4(𝔽113) generated by
| 70 | 91 | 0 | 0 |
| 22 | 27 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 28 | 109 |
| 78 | 7 | 0 | 0 |
| 67 | 35 | 0 | 0 |
| 0 | 0 | 48 | 42 |
| 0 | 0 | 7 | 65 |
| 78 | 7 | 0 | 0 |
| 67 | 35 | 0 | 0 |
| 0 | 0 | 48 | 42 |
| 0 | 0 | 77 | 65 |
G:=sub<GL(4,GF(113))| [70,22,0,0,91,27,0,0,0,0,28,28,0,0,0,109],[78,67,0,0,7,35,0,0,0,0,48,7,0,0,42,65],[78,67,0,0,7,35,0,0,0,0,48,77,0,0,42,65] >;
C56.82D4 in GAP, Magma, Sage, TeX
C_{56}._{82}D_4 % in TeX
G:=Group("C56.82D4"); // GroupNames label
G:=SmallGroup(448,650);
// by ID
G=gap.SmallGroup(448,650);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,344,254,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^28*b^-1>;
// generators/relations