metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.22D4, C7⋊C8.19D4, (C2×D8).4D7, C4.22(D4×D7), (C8×Dic7)⋊6C2, (C14×D8).5C2, (C2×D4).63D14, C28.165(C2×D4), (C2×C8).237D14, C7⋊3(C8.12D4), C8.15(C7⋊D4), (C2×Dic28)⋊18C2, C14.33(C4○D8), C28.17D4⋊5C2, (C2×C56).89C22, C22.255(D4×D7), C2.17(D8⋊3D7), C14.28(C4⋊1D4), C2.19(C28⋊D4), (C2×C28).432C23, (C2×Dic7).111D4, (D4×C14).82C22, (C4×Dic7).239C22, (C2×Dic14).121C22, C4.6(C2×C7⋊D4), (C2×D4.D7)⋊18C2, (C2×C14).345(C2×D4), (C2×C7⋊C8).270C22, (C2×C4).522(C22×D7), SmallGroup(448,689)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.22D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a41, cac-1=a-1, cbc-1=a28b-1 >
Subgroups: 612 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, C8.12D4, Dic28, C2×C7⋊C8, C4×Dic7, D4.D7, C23.D7, C2×C56, C7×D8, C2×Dic14, D4×C14, C8×Dic7, C2×Dic28, C2×D4.D7, C28.17D4, C14×D8, C56.22D4
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, D8⋊3D7, C28⋊D4, C56.22D4
►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)
(1 222 162 87)(2 207 163 72)(3 192 164 57)(4 177 165 98)(5 218 166 83)(6 203 167 68)(7 188 168 109)(8 173 113 94)(9 214 114 79)(10 199 115 64)(11 184 116 105)(12 169 117 90)(13 210 118 75)(14 195 119 60)(15 180 120 101)(16 221 121 86)(17 206 122 71)(18 191 123 112)(19 176 124 97)(20 217 125 82)(21 202 126 67)(22 187 127 108)(23 172 128 93)(24 213 129 78)(25 198 130 63)(26 183 131 104)(27 224 132 89)(28 209 133 74)(29 194 134 59)(30 179 135 100)(31 220 136 85)(32 205 137 70)(33 190 138 111)(34 175 139 96)(35 216 140 81)(36 201 141 66)(37 186 142 107)(38 171 143 92)(39 212 144 77)(40 197 145 62)(41 182 146 103)(42 223 147 88)(43 208 148 73)(44 193 149 58)(45 178 150 99)(46 219 151 84)(47 204 152 69)(48 189 153 110)(49 174 154 95)(50 215 155 80)(51 200 156 65)(52 185 157 106)(53 170 158 91)(54 211 159 76)(55 196 160 61)(56 181 161 102)
(1 59 29 87)(2 58 30 86)(3 57 31 85)(4 112 32 84)(5 111 33 83)(6 110 34 82)(7 109 35 81)(8 108 36 80)(9 107 37 79)(10 106 38 78)(11 105 39 77)(12 104 40 76)(13 103 41 75)(14 102 42 74)(15 101 43 73)(16 100 44 72)(17 99 45 71)(18 98 46 70)(19 97 47 69)(20 96 48 68)(21 95 49 67)(22 94 50 66)(23 93 51 65)(24 92 52 64)(25 91 53 63)(26 90 54 62)(27 89 55 61)(28 88 56 60)(113 187 141 215)(114 186 142 214)(115 185 143 213)(116 184 144 212)(117 183 145 211)(118 182 146 210)(119 181 147 209)(120 180 148 208)(121 179 149 207)(122 178 150 206)(123 177 151 205)(124 176 152 204)(125 175 153 203)(126 174 154 202)(127 173 155 201)(128 172 156 200)(129 171 157 199)(130 170 158 198)(131 169 159 197)(132 224 160 196)(133 223 161 195)(134 222 162 194)(135 221 163 193)(136 220 164 192)(137 219 165 191)(138 218 166 190)(139 217 167 189)(140 216 168 188)
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), (1,222,162,87)(2,207,163,72)(3,192,164,57)(4,177,165,98)(5,218,166,83)(6,203,167,68)(7,188,168,109)(8,173,113,94)(9,214,114,79)(10,199,115,64)(11,184,116,105)(12,169,117,90)(13,210,118,75)(14,195,119,60)(15,180,120,101)(16,221,121,86)(17,206,122,71)(18,191,123,112)(19,176,124,97)(20,217,125,82)(21,202,126,67)(22,187,127,108)(23,172,128,93)(24,213,129,78)(25,198,130,63)(26,183,131,104)(27,224,132,89)(28,209,133,74)(29,194,134,59)(30,179,135,100)(31,220,136,85)(32,205,137,70)(33,190,138,111)(34,175,139,96)(35,216,140,81)(36,201,141,66)(37,186,142,107)(38,171,143,92)(39,212,144,77)(40,197,145,62)(41,182,146,103)(42,223,147,88)(43,208,148,73)(44,193,149,58)(45,178,150,99)(46,219,151,84)(47,204,152,69)(48,189,153,110)(49,174,154,95)(50,215,155,80)(51,200,156,65)(52,185,157,106)(53,170,158,91)(54,211,159,76)(55,196,160,61)(56,181,161,102), (1,59,29,87)(2,58,30,86)(3,57,31,85)(4,112,32,84)(5,111,33,83)(6,110,34,82)(7,109,35,81)(8,108,36,80)(9,107,37,79)(10,106,38,78)(11,105,39,77)(12,104,40,76)(13,103,41,75)(14,102,42,74)(15,101,43,73)(16,100,44,72)(17,99,45,71)(18,98,46,70)(19,97,47,69)(20,96,48,68)(21,95,49,67)(22,94,50,66)(23,93,51,65)(24,92,52,64)(25,91,53,63)(26,90,54,62)(27,89,55,61)(28,88,56,60)(113,187,141,215)(114,186,142,214)(115,185,143,213)(116,184,144,212)(117,183,145,211)(118,182,146,210)(119,181,147,209)(120,180,148,208)(121,179,149,207)(122,178,150,206)(123,177,151,205)(124,176,152,204)(125,175,153,203)(126,174,154,202)(127,173,155,201)(128,172,156,200)(129,171,157,199)(130,170,158,198)(131,169,159,197)(132,224,160,196)(133,223,161,195)(134,222,162,194)(135,221,163,193)(136,220,164,192)(137,219,165,191)(138,218,166,190)(139,217,167,189)(140,216,168,188)>;
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), (1,222,162,87)(2,207,163,72)(3,192,164,57)(4,177,165,98)(5,218,166,83)(6,203,167,68)(7,188,168,109)(8,173,113,94)(9,214,114,79)(10,199,115,64)(11,184,116,105)(12,169,117,90)(13,210,118,75)(14,195,119,60)(15,180,120,101)(16,221,121,86)(17,206,122,71)(18,191,123,112)(19,176,124,97)(20,217,125,82)(21,202,126,67)(22,187,127,108)(23,172,128,93)(24,213,129,78)(25,198,130,63)(26,183,131,104)(27,224,132,89)(28,209,133,74)(29,194,134,59)(30,179,135,100)(31,220,136,85)(32,205,137,70)(33,190,138,111)(34,175,139,96)(35,216,140,81)(36,201,141,66)(37,186,142,107)(38,171,143,92)(39,212,144,77)(40,197,145,62)(41,182,146,103)(42,223,147,88)(43,208,148,73)(44,193,149,58)(45,178,150,99)(46,219,151,84)(47,204,152,69)(48,189,153,110)(49,174,154,95)(50,215,155,80)(51,200,156,65)(52,185,157,106)(53,170,158,91)(54,211,159,76)(55,196,160,61)(56,181,161,102), (1,59,29,87)(2,58,30,86)(3,57,31,85)(4,112,32,84)(5,111,33,83)(6,110,34,82)(7,109,35,81)(8,108,36,80)(9,107,37,79)(10,106,38,78)(11,105,39,77)(12,104,40,76)(13,103,41,75)(14,102,42,74)(15,101,43,73)(16,100,44,72)(17,99,45,71)(18,98,46,70)(19,97,47,69)(20,96,48,68)(21,95,49,67)(22,94,50,66)(23,93,51,65)(24,92,52,64)(25,91,53,63)(26,90,54,62)(27,89,55,61)(28,88,56,60)(113,187,141,215)(114,186,142,214)(115,185,143,213)(116,184,144,212)(117,183,145,211)(118,182,146,210)(119,181,147,209)(120,180,148,208)(121,179,149,207)(122,178,150,206)(123,177,151,205)(124,176,152,204)(125,175,153,203)(126,174,154,202)(127,173,155,201)(128,172,156,200)(129,171,157,199)(130,170,158,198)(131,169,159,197)(132,224,160,196)(133,223,161,195)(134,222,162,194)(135,221,163,193)(136,220,164,192)(137,219,165,191)(138,218,166,190)(139,217,167,189)(140,216,168,188) );
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)], [(1,222,162,87),(2,207,163,72),(3,192,164,57),(4,177,165,98),(5,218,166,83),(6,203,167,68),(7,188,168,109),(8,173,113,94),(9,214,114,79),(10,199,115,64),(11,184,116,105),(12,169,117,90),(13,210,118,75),(14,195,119,60),(15,180,120,101),(16,221,121,86),(17,206,122,71),(18,191,123,112),(19,176,124,97),(20,217,125,82),(21,202,126,67),(22,187,127,108),(23,172,128,93),(24,213,129,78),(25,198,130,63),(26,183,131,104),(27,224,132,89),(28,209,133,74),(29,194,134,59),(30,179,135,100),(31,220,136,85),(32,205,137,70),(33,190,138,111),(34,175,139,96),(35,216,140,81),(36,201,141,66),(37,186,142,107),(38,171,143,92),(39,212,144,77),(40,197,145,62),(41,182,146,103),(42,223,147,88),(43,208,148,73),(44,193,149,58),(45,178,150,99),(46,219,151,84),(47,204,152,69),(48,189,153,110),(49,174,154,95),(50,215,155,80),(51,200,156,65),(52,185,157,106),(53,170,158,91),(54,211,159,76),(55,196,160,61),(56,181,161,102)], [(1,59,29,87),(2,58,30,86),(3,57,31,85),(4,112,32,84),(5,111,33,83),(6,110,34,82),(7,109,35,81),(8,108,36,80),(9,107,37,79),(10,106,38,78),(11,105,39,77),(12,104,40,76),(13,103,41,75),(14,102,42,74),(15,101,43,73),(16,100,44,72),(17,99,45,71),(18,98,46,70),(19,97,47,69),(20,96,48,68),(21,95,49,67),(22,94,50,66),(23,93,51,65),(24,92,52,64),(25,91,53,63),(26,90,54,62),(27,89,55,61),(28,88,56,60),(113,187,141,215),(114,186,142,214),(115,185,143,213),(116,184,144,212),(117,183,145,211),(118,182,146,210),(119,181,147,209),(120,180,148,208),(121,179,149,207),(122,178,150,206),(123,177,151,205),(124,176,152,204),(125,175,153,203),(126,174,154,202),(127,173,155,201),(128,172,156,200),(129,171,157,199),(130,170,158,198),(131,169,159,197),(132,224,160,196),(133,223,161,195),(134,222,162,194),(135,221,163,193),(136,220,164,192),(137,219,165,191),(138,218,166,190),(139,217,167,189),(140,216,168,188)]])
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 | ··· | 14U | 28A | ··· | 28F | 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 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 14 | 14 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | C4○D8 | C7⋊D4 | D4×D7 | D4×D7 | D8⋊3D7 |
| kernel | C56.22D4 | C8×Dic7 | C2×Dic28 | C2×D4.D7 | C28.17D4 | C14×D8 | C7⋊C8 | C56 | C2×Dic7 | C2×D8 | C2×C8 | C2×D4 | C14 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 8 | 12 | 3 | 3 | 12 |
Matrix representation of C56.22D4 ►in GL4(𝔽113) generated by
| 97 | 17 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 15 | 62 |
| 1 | 28 | 0 | 0 |
| 8 | 112 | 0 | 0 |
| 0 | 0 | 15 | 62 |
| 0 | 0 | 62 | 98 |
| 112 | 0 | 0 | 0 |
| 105 | 1 | 0 | 0 |
| 0 | 0 | 15 | 62 |
| 0 | 0 | 0 | 98 |
G:=sub<GL(4,GF(113))| [97,0,0,0,17,7,0,0,0,0,0,15,0,0,15,62],[1,8,0,0,28,112,0,0,0,0,15,62,0,0,62,98],[112,105,0,0,0,1,0,0,0,0,15,0,0,0,62,98] >;
C56.22D4 in GAP, Magma, Sage, TeX
C_{56}._{22}D_4 % in TeX
G:=Group("C56.22D4"); // GroupNames label
G:=SmallGroup(448,689);
// by ID
G=gap.SmallGroup(448,689);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,1094,135,570,297,136,18822]);
// 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^-1,c*b*c^-1=a^28*b^-1>;
// generators/relations