metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.8D4, C8.18D28, C28.49D8, Dic28.4C4, C8.3(C4×D7), C56.23(C2×C4), (C2×C8).44D14, (C2×C28).96D4, C4.22(D4⋊D7), C4.5(D14⋊C4), C7⋊1(C8.17D4), (C2×C14).5SD16, C8.C4.3D7, C28.C8.3C2, C28.5(C22⋊C4), C22.4(Q8⋊D7), C14.8(D4⋊C4), (C2×C56).101C22, (C2×Dic28).12C2, C2.10(C14.D8), (C7×C8.C4).2C2, (C2×C4).19(C7⋊D4), SmallGroup(448,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.8D4
G = < a,b,c | a56=1, b4=a28, c2=a21, bab-1=a15, cac-1=a41, cbc-1=a21b3 >
Smallest permutation representation of C56.8D4
►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 61 15 103 29 89 43 75)(2 76 16 62 30 104 44 90)(3 91 17 77 31 63 45 105)(4 106 18 92 32 78 46 64)(5 65 19 107 33 93 47 79)(6 80 20 66 34 108 48 94)(7 95 21 81 35 67 49 109)(8 110 22 96 36 82 50 68)(9 69 23 111 37 97 51 83)(10 84 24 70 38 112 52 98)(11 99 25 85 39 71 53 57)(12 58 26 100 40 86 54 72)(13 73 27 59 41 101 55 87)(14 88 28 74 42 60 56 102)(113 183 127 169 141 211 155 197)(114 198 128 184 142 170 156 212)(115 213 129 199 143 185 157 171)(116 172 130 214 144 200 158 186)(117 187 131 173 145 215 159 201)(118 202 132 188 146 174 160 216)(119 217 133 203 147 189 161 175)(120 176 134 218 148 204 162 190)(121 191 135 177 149 219 163 205)(122 206 136 192 150 178 164 220)(123 221 137 207 151 193 165 179)(124 180 138 222 152 208 166 194)(125 195 139 181 153 223 167 209)(126 210 140 196 154 182 168 224)
(1 197 22 218 43 183 8 204 29 169 50 190 15 211 36 176)(2 182 23 203 44 224 9 189 30 210 51 175 16 196 37 217)(3 223 24 188 45 209 10 174 31 195 52 216 17 181 38 202)(4 208 25 173 46 194 11 215 32 180 53 201 18 222 39 187)(5 193 26 214 47 179 12 200 33 221 54 186 19 207 40 172)(6 178 27 199 48 220 13 185 34 206 55 171 20 192 41 213)(7 219 28 184 49 205 14 170 35 191 56 212 21 177 42 198)(57 152 78 117 99 138 64 159 85 124 106 145 71 166 92 131)(58 137 79 158 100 123 65 144 86 165 107 130 72 151 93 116)(59 122 80 143 101 164 66 129 87 150 108 115 73 136 94 157)(60 163 81 128 102 149 67 114 88 135 109 156 74 121 95 142)(61 148 82 113 103 134 68 155 89 120 110 141 75 162 96 127)(62 133 83 154 104 119 69 140 90 161 111 126 76 147 97 168)(63 118 84 139 105 160 70 125 91 146 112 167 77 132 98 153)
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), (1,61,15,103,29,89,43,75)(2,76,16,62,30,104,44,90)(3,91,17,77,31,63,45,105)(4,106,18,92,32,78,46,64)(5,65,19,107,33,93,47,79)(6,80,20,66,34,108,48,94)(7,95,21,81,35,67,49,109)(8,110,22,96,36,82,50,68)(9,69,23,111,37,97,51,83)(10,84,24,70,38,112,52,98)(11,99,25,85,39,71,53,57)(12,58,26,100,40,86,54,72)(13,73,27,59,41,101,55,87)(14,88,28,74,42,60,56,102)(113,183,127,169,141,211,155,197)(114,198,128,184,142,170,156,212)(115,213,129,199,143,185,157,171)(116,172,130,214,144,200,158,186)(117,187,131,173,145,215,159,201)(118,202,132,188,146,174,160,216)(119,217,133,203,147,189,161,175)(120,176,134,218,148,204,162,190)(121,191,135,177,149,219,163,205)(122,206,136,192,150,178,164,220)(123,221,137,207,151,193,165,179)(124,180,138,222,152,208,166,194)(125,195,139,181,153,223,167,209)(126,210,140,196,154,182,168,224), (1,197,22,218,43,183,8,204,29,169,50,190,15,211,36,176)(2,182,23,203,44,224,9,189,30,210,51,175,16,196,37,217)(3,223,24,188,45,209,10,174,31,195,52,216,17,181,38,202)(4,208,25,173,46,194,11,215,32,180,53,201,18,222,39,187)(5,193,26,214,47,179,12,200,33,221,54,186,19,207,40,172)(6,178,27,199,48,220,13,185,34,206,55,171,20,192,41,213)(7,219,28,184,49,205,14,170,35,191,56,212,21,177,42,198)(57,152,78,117,99,138,64,159,85,124,106,145,71,166,92,131)(58,137,79,158,100,123,65,144,86,165,107,130,72,151,93,116)(59,122,80,143,101,164,66,129,87,150,108,115,73,136,94,157)(60,163,81,128,102,149,67,114,88,135,109,156,74,121,95,142)(61,148,82,113,103,134,68,155,89,120,110,141,75,162,96,127)(62,133,83,154,104,119,69,140,90,161,111,126,76,147,97,168)(63,118,84,139,105,160,70,125,91,146,112,167,77,132,98,153)>;
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), (1,61,15,103,29,89,43,75)(2,76,16,62,30,104,44,90)(3,91,17,77,31,63,45,105)(4,106,18,92,32,78,46,64)(5,65,19,107,33,93,47,79)(6,80,20,66,34,108,48,94)(7,95,21,81,35,67,49,109)(8,110,22,96,36,82,50,68)(9,69,23,111,37,97,51,83)(10,84,24,70,38,112,52,98)(11,99,25,85,39,71,53,57)(12,58,26,100,40,86,54,72)(13,73,27,59,41,101,55,87)(14,88,28,74,42,60,56,102)(113,183,127,169,141,211,155,197)(114,198,128,184,142,170,156,212)(115,213,129,199,143,185,157,171)(116,172,130,214,144,200,158,186)(117,187,131,173,145,215,159,201)(118,202,132,188,146,174,160,216)(119,217,133,203,147,189,161,175)(120,176,134,218,148,204,162,190)(121,191,135,177,149,219,163,205)(122,206,136,192,150,178,164,220)(123,221,137,207,151,193,165,179)(124,180,138,222,152,208,166,194)(125,195,139,181,153,223,167,209)(126,210,140,196,154,182,168,224), (1,197,22,218,43,183,8,204,29,169,50,190,15,211,36,176)(2,182,23,203,44,224,9,189,30,210,51,175,16,196,37,217)(3,223,24,188,45,209,10,174,31,195,52,216,17,181,38,202)(4,208,25,173,46,194,11,215,32,180,53,201,18,222,39,187)(5,193,26,214,47,179,12,200,33,221,54,186,19,207,40,172)(6,178,27,199,48,220,13,185,34,206,55,171,20,192,41,213)(7,219,28,184,49,205,14,170,35,191,56,212,21,177,42,198)(57,152,78,117,99,138,64,159,85,124,106,145,71,166,92,131)(58,137,79,158,100,123,65,144,86,165,107,130,72,151,93,116)(59,122,80,143,101,164,66,129,87,150,108,115,73,136,94,157)(60,163,81,128,102,149,67,114,88,135,109,156,74,121,95,142)(61,148,82,113,103,134,68,155,89,120,110,141,75,162,96,127)(62,133,83,154,104,119,69,140,90,161,111,126,76,147,97,168)(63,118,84,139,105,160,70,125,91,146,112,167,77,132,98,153) );
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)], [(1,61,15,103,29,89,43,75),(2,76,16,62,30,104,44,90),(3,91,17,77,31,63,45,105),(4,106,18,92,32,78,46,64),(5,65,19,107,33,93,47,79),(6,80,20,66,34,108,48,94),(7,95,21,81,35,67,49,109),(8,110,22,96,36,82,50,68),(9,69,23,111,37,97,51,83),(10,84,24,70,38,112,52,98),(11,99,25,85,39,71,53,57),(12,58,26,100,40,86,54,72),(13,73,27,59,41,101,55,87),(14,88,28,74,42,60,56,102),(113,183,127,169,141,211,155,197),(114,198,128,184,142,170,156,212),(115,213,129,199,143,185,157,171),(116,172,130,214,144,200,158,186),(117,187,131,173,145,215,159,201),(118,202,132,188,146,174,160,216),(119,217,133,203,147,189,161,175),(120,176,134,218,148,204,162,190),(121,191,135,177,149,219,163,205),(122,206,136,192,150,178,164,220),(123,221,137,207,151,193,165,179),(124,180,138,222,152,208,166,194),(125,195,139,181,153,223,167,209),(126,210,140,196,154,182,168,224)], [(1,197,22,218,43,183,8,204,29,169,50,190,15,211,36,176),(2,182,23,203,44,224,9,189,30,210,51,175,16,196,37,217),(3,223,24,188,45,209,10,174,31,195,52,216,17,181,38,202),(4,208,25,173,46,194,11,215,32,180,53,201,18,222,39,187),(5,193,26,214,47,179,12,200,33,221,54,186,19,207,40,172),(6,178,27,199,48,220,13,185,34,206,55,171,20,192,41,213),(7,219,28,184,49,205,14,170,35,191,56,212,21,177,42,198),(57,152,78,117,99,138,64,159,85,124,106,145,71,166,92,131),(58,137,79,158,100,123,65,144,86,165,107,130,72,151,93,116),(59,122,80,143,101,164,66,129,87,150,108,115,73,136,94,157),(60,163,81,128,102,149,67,114,88,135,109,156,74,121,95,142),(61,148,82,113,103,134,68,155,89,120,110,141,75,162,96,127),(62,133,83,154,104,119,69,140,90,161,111,126,76,147,97,168),(63,118,84,139,105,160,70,125,91,146,112,167,77,132,98,153)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | D28 | C7⋊D4 | C8.17D4 | D4⋊D7 | Q8⋊D7 | C56.8D4 |
| kernel | C56.8D4 | C28.C8 | C7×C8.C4 | C2×Dic28 | Dic28 | C56 | C2×C28 | C8.C4 | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of C56.8D4 ►in GL4(𝔽113) generated by
| 78 | 84 | 0 | 0 |
| 58 | 90 | 0 | 0 |
| 76 | 3 | 42 | 91 |
| 107 | 28 | 44 | 55 |
| 19 | 75 | 111 | 0 |
| 76 | 62 | 0 | 111 |
| 107 | 66 | 94 | 38 |
| 94 | 68 | 37 | 51 |
| 87 | 108 | 11 | 47 |
| 67 | 26 | 31 | 102 |
| 14 | 74 | 24 | 34 |
| 37 | 99 | 23 | 89 |
G:=sub<GL(4,GF(113))| [78,58,76,107,84,90,3,28,0,0,42,44,0,0,91,55],[19,76,107,94,75,62,66,68,111,0,94,37,0,111,38,51],[87,67,14,37,108,26,74,99,11,31,24,23,47,102,34,89] >;
C56.8D4 in GAP, Magma, Sage, TeX
C_{56}._8D_4 % in TeX
G:=Group("C56.8D4"); // GroupNames label
G:=SmallGroup(448,53);
// by ID
G=gap.SmallGroup(448,53);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,758,184,675,794,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=a^28,c^2=a^21,b*a*b^-1=a^15,c*a*c^-1=a^41,c*b*c^-1=a^21*b^3>;
// generators/relations
Export