metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.1D4, C8.1D28, C42.21D14, C8⋊C4⋊4D7, C28⋊2Q8⋊4C2, (C2×C4).28D28, (C2×C28).39D4, C4.35(C2×D28), (C2×C8).56D14, C7⋊1(C8.2D4), C28.278(C2×D4), (C4×C28).6C22, (C2×Dic28)⋊10C2, C14.8(C4⋊1D4), (C2×C56).57C22, C4.D28.3C2, (C2×D28).9C22, C2.10(C28⋊4D4), (C2×C28).736C23, C2.9(C8.D14), C22.100(C2×D28), C14.5(C8.C22), (C2×Dic14).9C22, (C7×C8⋊C4)⋊5C2, (C2×C56⋊C2).2C2, (C2×C14).119(C2×D4), (C2×C4).680(C22×D7), SmallGroup(448,249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D28
G = < a,b,c | a8=b28=1, c2=a4, bab-1=a5, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 772 in 124 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C56, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C8.2D4, C56⋊C2, Dic28, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×Dic14, C2×D28, C7×C8⋊C4, C28⋊2Q8, C4.D28, C2×C56⋊C2, C2×Dic28, C8.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8.C22, D28, C22×D7, C8.2D4, C2×D28, C28⋊4D4, C8.D14, C8.D28
►On 224 points
Generators in S224
(1 62 137 98 29 200 178 159)(2 201 138 160 30 63 179 99)(3 64 139 100 31 202 180 161)(4 203 140 162 32 65 181 101)(5 66 113 102 33 204 182 163)(6 205 114 164 34 67 183 103)(7 68 115 104 35 206 184 165)(8 207 116 166 36 69 185 105)(9 70 117 106 37 208 186 167)(10 209 118 168 38 71 187 107)(11 72 119 108 39 210 188 141)(12 211 120 142 40 73 189 109)(13 74 121 110 41 212 190 143)(14 213 122 144 42 75 191 111)(15 76 123 112 43 214 192 145)(16 215 124 146 44 77 193 85)(17 78 125 86 45 216 194 147)(18 217 126 148 46 79 195 87)(19 80 127 88 47 218 196 149)(20 219 128 150 48 81 169 89)(21 82 129 90 49 220 170 151)(22 221 130 152 50 83 171 91)(23 84 131 92 51 222 172 153)(24 223 132 154 52 57 173 93)(25 58 133 94 53 224 174 155)(26 197 134 156 54 59 175 95)(27 60 135 96 55 198 176 157)(28 199 136 158 56 61 177 97)
(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 56 29 28)(2 27 30 55)(3 54 31 26)(4 25 32 53)(5 52 33 24)(6 23 34 51)(7 50 35 22)(8 21 36 49)(9 48 37 20)(10 19 38 47)(11 46 39 18)(12 17 40 45)(13 44 41 16)(14 15 42 43)(57 102 223 163)(58 162 224 101)(59 100 197 161)(60 160 198 99)(61 98 199 159)(62 158 200 97)(63 96 201 157)(64 156 202 95)(65 94 203 155)(66 154 204 93)(67 92 205 153)(68 152 206 91)(69 90 207 151)(70 150 208 89)(71 88 209 149)(72 148 210 87)(73 86 211 147)(74 146 212 85)(75 112 213 145)(76 144 214 111)(77 110 215 143)(78 142 216 109)(79 108 217 141)(80 168 218 107)(81 106 219 167)(82 166 220 105)(83 104 221 165)(84 164 222 103)(113 132 182 173)(114 172 183 131)(115 130 184 171)(116 170 185 129)(117 128 186 169)(118 196 187 127)(119 126 188 195)(120 194 189 125)(121 124 190 193)(122 192 191 123)(133 140 174 181)(134 180 175 139)(135 138 176 179)(136 178 177 137)
G:=sub<Sym(224)| (1,62,137,98,29,200,178,159)(2,201,138,160,30,63,179,99)(3,64,139,100,31,202,180,161)(4,203,140,162,32,65,181,101)(5,66,113,102,33,204,182,163)(6,205,114,164,34,67,183,103)(7,68,115,104,35,206,184,165)(8,207,116,166,36,69,185,105)(9,70,117,106,37,208,186,167)(10,209,118,168,38,71,187,107)(11,72,119,108,39,210,188,141)(12,211,120,142,40,73,189,109)(13,74,121,110,41,212,190,143)(14,213,122,144,42,75,191,111)(15,76,123,112,43,214,192,145)(16,215,124,146,44,77,193,85)(17,78,125,86,45,216,194,147)(18,217,126,148,46,79,195,87)(19,80,127,88,47,218,196,149)(20,219,128,150,48,81,169,89)(21,82,129,90,49,220,170,151)(22,221,130,152,50,83,171,91)(23,84,131,92,51,222,172,153)(24,223,132,154,52,57,173,93)(25,58,133,94,53,224,174,155)(26,197,134,156,54,59,175,95)(27,60,135,96,55,198,176,157)(28,199,136,158,56,61,177,97), (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), (1,56,29,28)(2,27,30,55)(3,54,31,26)(4,25,32,53)(5,52,33,24)(6,23,34,51)(7,50,35,22)(8,21,36,49)(9,48,37,20)(10,19,38,47)(11,46,39,18)(12,17,40,45)(13,44,41,16)(14,15,42,43)(57,102,223,163)(58,162,224,101)(59,100,197,161)(60,160,198,99)(61,98,199,159)(62,158,200,97)(63,96,201,157)(64,156,202,95)(65,94,203,155)(66,154,204,93)(67,92,205,153)(68,152,206,91)(69,90,207,151)(70,150,208,89)(71,88,209,149)(72,148,210,87)(73,86,211,147)(74,146,212,85)(75,112,213,145)(76,144,214,111)(77,110,215,143)(78,142,216,109)(79,108,217,141)(80,168,218,107)(81,106,219,167)(82,166,220,105)(83,104,221,165)(84,164,222,103)(113,132,182,173)(114,172,183,131)(115,130,184,171)(116,170,185,129)(117,128,186,169)(118,196,187,127)(119,126,188,195)(120,194,189,125)(121,124,190,193)(122,192,191,123)(133,140,174,181)(134,180,175,139)(135,138,176,179)(136,178,177,137)>;
G:=Group( (1,62,137,98,29,200,178,159)(2,201,138,160,30,63,179,99)(3,64,139,100,31,202,180,161)(4,203,140,162,32,65,181,101)(5,66,113,102,33,204,182,163)(6,205,114,164,34,67,183,103)(7,68,115,104,35,206,184,165)(8,207,116,166,36,69,185,105)(9,70,117,106,37,208,186,167)(10,209,118,168,38,71,187,107)(11,72,119,108,39,210,188,141)(12,211,120,142,40,73,189,109)(13,74,121,110,41,212,190,143)(14,213,122,144,42,75,191,111)(15,76,123,112,43,214,192,145)(16,215,124,146,44,77,193,85)(17,78,125,86,45,216,194,147)(18,217,126,148,46,79,195,87)(19,80,127,88,47,218,196,149)(20,219,128,150,48,81,169,89)(21,82,129,90,49,220,170,151)(22,221,130,152,50,83,171,91)(23,84,131,92,51,222,172,153)(24,223,132,154,52,57,173,93)(25,58,133,94,53,224,174,155)(26,197,134,156,54,59,175,95)(27,60,135,96,55,198,176,157)(28,199,136,158,56,61,177,97), (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), (1,56,29,28)(2,27,30,55)(3,54,31,26)(4,25,32,53)(5,52,33,24)(6,23,34,51)(7,50,35,22)(8,21,36,49)(9,48,37,20)(10,19,38,47)(11,46,39,18)(12,17,40,45)(13,44,41,16)(14,15,42,43)(57,102,223,163)(58,162,224,101)(59,100,197,161)(60,160,198,99)(61,98,199,159)(62,158,200,97)(63,96,201,157)(64,156,202,95)(65,94,203,155)(66,154,204,93)(67,92,205,153)(68,152,206,91)(69,90,207,151)(70,150,208,89)(71,88,209,149)(72,148,210,87)(73,86,211,147)(74,146,212,85)(75,112,213,145)(76,144,214,111)(77,110,215,143)(78,142,216,109)(79,108,217,141)(80,168,218,107)(81,106,219,167)(82,166,220,105)(83,104,221,165)(84,164,222,103)(113,132,182,173)(114,172,183,131)(115,130,184,171)(116,170,185,129)(117,128,186,169)(118,196,187,127)(119,126,188,195)(120,194,189,125)(121,124,190,193)(122,192,191,123)(133,140,174,181)(134,180,175,139)(135,138,176,179)(136,178,177,137) );
G=PermutationGroup([[(1,62,137,98,29,200,178,159),(2,201,138,160,30,63,179,99),(3,64,139,100,31,202,180,161),(4,203,140,162,32,65,181,101),(5,66,113,102,33,204,182,163),(6,205,114,164,34,67,183,103),(7,68,115,104,35,206,184,165),(8,207,116,166,36,69,185,105),(9,70,117,106,37,208,186,167),(10,209,118,168,38,71,187,107),(11,72,119,108,39,210,188,141),(12,211,120,142,40,73,189,109),(13,74,121,110,41,212,190,143),(14,213,122,144,42,75,191,111),(15,76,123,112,43,214,192,145),(16,215,124,146,44,77,193,85),(17,78,125,86,45,216,194,147),(18,217,126,148,46,79,195,87),(19,80,127,88,47,218,196,149),(20,219,128,150,48,81,169,89),(21,82,129,90,49,220,170,151),(22,221,130,152,50,83,171,91),(23,84,131,92,51,222,172,153),(24,223,132,154,52,57,173,93),(25,58,133,94,53,224,174,155),(26,197,134,156,54,59,175,95),(27,60,135,96,55,198,176,157),(28,199,136,158,56,61,177,97)], [(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)], [(1,56,29,28),(2,27,30,55),(3,54,31,26),(4,25,32,53),(5,52,33,24),(6,23,34,51),(7,50,35,22),(8,21,36,49),(9,48,37,20),(10,19,38,47),(11,46,39,18),(12,17,40,45),(13,44,41,16),(14,15,42,43),(57,102,223,163),(58,162,224,101),(59,100,197,161),(60,160,198,99),(61,98,199,159),(62,158,200,97),(63,96,201,157),(64,156,202,95),(65,94,203,155),(66,154,204,93),(67,92,205,153),(68,152,206,91),(69,90,207,151),(70,150,208,89),(71,88,209,149),(72,148,210,87),(73,86,211,147),(74,146,212,85),(75,112,213,145),(76,144,214,111),(77,110,215,143),(78,142,216,109),(79,108,217,141),(80,168,218,107),(81,106,219,167),(82,166,220,105),(83,104,221,165),(84,164,222,103),(113,132,182,173),(114,172,183,131),(115,130,184,171),(116,170,185,129),(117,128,186,169),(118,196,187,127),(119,126,188,195),(120,194,189,125),(121,124,190,193),(122,192,191,123),(133,140,174,181),(134,180,175,139),(135,138,176,179),(136,178,177,137)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 2 | 2 | 4 | 4 | 56 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D28 | D28 | C8.C22 | C8.D14 |
| kernel | C8.D28 | C7×C8⋊C4 | C28⋊2Q8 | C4.D28 | C2×C56⋊C2 | C2×Dic28 | C56 | C2×C28 | C8⋊C4 | C42 | C2×C8 | C8 | C2×C4 | C14 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 3 | 3 | 6 | 24 | 12 | 2 | 12 |
Matrix representation of C8.D28 ►in GL6(𝔽113)
| 112 | 111 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 71 | 92 | 0 |
| 0 | 0 | 42 | 63 | 0 | 92 |
| 0 | 0 | 18 | 100 | 59 | 42 |
| 0 | 0 | 13 | 45 | 71 | 50 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 22 | 99 | 31 |
| 0 | 0 | 91 | 94 | 82 | 52 |
| 0 | 0 | 64 | 49 | 95 | 91 |
| 0 | 0 | 64 | 110 | 22 | 19 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 22 | 99 | 31 |
| 0 | 0 | 2 | 95 | 34 | 14 |
| 0 | 0 | 60 | 77 | 65 | 39 |
| 0 | 0 | 48 | 53 | 17 | 48 |
G:=sub<GL(6,GF(113))| [112,1,0,0,0,0,111,1,0,0,0,0,0,0,54,42,18,13,0,0,71,63,100,45,0,0,92,0,59,71,0,0,0,92,42,50],[1,112,0,0,0,0,2,112,0,0,0,0,0,0,18,91,64,64,0,0,22,94,49,110,0,0,99,82,95,22,0,0,31,52,91,19],[1,0,0,0,0,0,2,112,0,0,0,0,0,0,18,2,60,48,0,0,22,95,77,53,0,0,99,34,65,17,0,0,31,14,39,48] >;
C8.D28 in GAP, Magma, Sage, TeX
C_8.D_{28} % in TeX
G:=Group("C8.D28"); // GroupNames label
G:=SmallGroup(448,249);
// by ID
G=gap.SmallGroup(448,249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,387,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations