direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8×D28, C56⋊24D4, C42.256D14, C7⋊1(C8×D4), C4⋊2(C8×D7), (C4×C8)⋊3D7, C28⋊4(C2×C8), (C4×C56)⋊19C2, D14⋊1(C2×C8), C14.4(C4×D4), C2.1(C4×D28), D14⋊C8⋊41C2, C28⋊C8⋊36C2, C4.74(C2×D28), D14⋊C4.18C4, (C2×D28).18C4, (C4×D28).29C2, C14.3(C8○D4), C28.294(C2×D4), (C2×C8).339D14, C4⋊Dic7.23C4, C14.4(C22×C8), C28.241(C4○D4), C4.125(C4○D28), (C2×C56).342C22, (C4×C28).323C22, (C2×C28).804C23, C2.2(D28.2C4), C2.6(D7×C2×C8), (D7×C2×C8)⋊11C2, C22.36(C2×C4×D7), (C2×C4).103(C4×D7), (C2×C28).220(C2×C4), (C2×C7⋊C8).290C22, (C2×C4×D7).266C22, (C2×C14).59(C22×C4), (C2×Dic7).46(C2×C4), (C22×D7).31(C2×C4), (C2×C4).746(C22×D7), SmallGroup(448,220)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×D28
G = < a,b,c | a8=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 580 in 134 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, C28, D14, D14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C7⋊C8, C56, C56, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C8×D4, C8×D7, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C28⋊C8, D14⋊C8, C4×C56, C4×D28, D7×C2×C8, C8×D28
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D7, C2×C8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C22×C8, C8○D4, C4×D7, D28, C22×D7, C8×D4, C8×D7, C2×C4×D7, C2×D28, C4○D28, C4×D28, D7×C2×C8, D28.2C4, C8×D28
►On 224 points
Generators in S224
(1 210 100 167 132 194 42 75)(2 211 101 168 133 195 43 76)(3 212 102 141 134 196 44 77)(4 213 103 142 135 169 45 78)(5 214 104 143 136 170 46 79)(6 215 105 144 137 171 47 80)(7 216 106 145 138 172 48 81)(8 217 107 146 139 173 49 82)(9 218 108 147 140 174 50 83)(10 219 109 148 113 175 51 84)(11 220 110 149 114 176 52 57)(12 221 111 150 115 177 53 58)(13 222 112 151 116 178 54 59)(14 223 85 152 117 179 55 60)(15 224 86 153 118 180 56 61)(16 197 87 154 119 181 29 62)(17 198 88 155 120 182 30 63)(18 199 89 156 121 183 31 64)(19 200 90 157 122 184 32 65)(20 201 91 158 123 185 33 66)(21 202 92 159 124 186 34 67)(22 203 93 160 125 187 35 68)(23 204 94 161 126 188 36 69)(24 205 95 162 127 189 37 70)(25 206 96 163 128 190 38 71)(26 207 97 164 129 191 39 72)(27 208 98 165 130 192 40 73)(28 209 99 166 131 193 41 74)
(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)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 130)(10 129)(11 128)(12 127)(13 126)(14 125)(15 124)(16 123)(17 122)(18 121)(19 120)(20 119)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 140)(28 139)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 163)(58 162)(59 161)(60 160)(61 159)(62 158)(63 157)(64 156)(65 155)(66 154)(67 153)(68 152)(69 151)(70 150)(71 149)(72 148)(73 147)(74 146)(75 145)(76 144)(77 143)(78 142)(79 141)(80 168)(81 167)(82 166)(83 165)(84 164)(169 213)(170 212)(171 211)(172 210)(173 209)(174 208)(175 207)(176 206)(177 205)(178 204)(179 203)(180 202)(181 201)(182 200)(183 199)(184 198)(185 197)(186 224)(187 223)(188 222)(189 221)(190 220)(191 219)(192 218)(193 217)(194 216)(195 215)(196 214)
G:=sub<Sym(224)| (1,210,100,167,132,194,42,75)(2,211,101,168,133,195,43,76)(3,212,102,141,134,196,44,77)(4,213,103,142,135,169,45,78)(5,214,104,143,136,170,46,79)(6,215,105,144,137,171,47,80)(7,216,106,145,138,172,48,81)(8,217,107,146,139,173,49,82)(9,218,108,147,140,174,50,83)(10,219,109,148,113,175,51,84)(11,220,110,149,114,176,52,57)(12,221,111,150,115,177,53,58)(13,222,112,151,116,178,54,59)(14,223,85,152,117,179,55,60)(15,224,86,153,118,180,56,61)(16,197,87,154,119,181,29,62)(17,198,88,155,120,182,30,63)(18,199,89,156,121,183,31,64)(19,200,90,157,122,184,32,65)(20,201,91,158,123,185,33,66)(21,202,92,159,124,186,34,67)(22,203,93,160,125,187,35,68)(23,204,94,161,126,188,36,69)(24,205,95,162,127,189,37,70)(25,206,96,163,128,190,38,71)(26,207,97,164,129,191,39,72)(27,208,98,165,130,192,40,73)(28,209,99,166,131,193,41,74), (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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,130)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,140)(28,139)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,163)(58,162)(59,161)(60,160)(61,159)(62,158)(63,157)(64,156)(65,155)(66,154)(67,153)(68,152)(69,151)(70,150)(71,149)(72,148)(73,147)(74,146)(75,145)(76,144)(77,143)(78,142)(79,141)(80,168)(81,167)(82,166)(83,165)(84,164)(169,213)(170,212)(171,211)(172,210)(173,209)(174,208)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,201)(182,200)(183,199)(184,198)(185,197)(186,224)(187,223)(188,222)(189,221)(190,220)(191,219)(192,218)(193,217)(194,216)(195,215)(196,214)>;
G:=Group( (1,210,100,167,132,194,42,75)(2,211,101,168,133,195,43,76)(3,212,102,141,134,196,44,77)(4,213,103,142,135,169,45,78)(5,214,104,143,136,170,46,79)(6,215,105,144,137,171,47,80)(7,216,106,145,138,172,48,81)(8,217,107,146,139,173,49,82)(9,218,108,147,140,174,50,83)(10,219,109,148,113,175,51,84)(11,220,110,149,114,176,52,57)(12,221,111,150,115,177,53,58)(13,222,112,151,116,178,54,59)(14,223,85,152,117,179,55,60)(15,224,86,153,118,180,56,61)(16,197,87,154,119,181,29,62)(17,198,88,155,120,182,30,63)(18,199,89,156,121,183,31,64)(19,200,90,157,122,184,32,65)(20,201,91,158,123,185,33,66)(21,202,92,159,124,186,34,67)(22,203,93,160,125,187,35,68)(23,204,94,161,126,188,36,69)(24,205,95,162,127,189,37,70)(25,206,96,163,128,190,38,71)(26,207,97,164,129,191,39,72)(27,208,98,165,130,192,40,73)(28,209,99,166,131,193,41,74), (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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,130)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,140)(28,139)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,163)(58,162)(59,161)(60,160)(61,159)(62,158)(63,157)(64,156)(65,155)(66,154)(67,153)(68,152)(69,151)(70,150)(71,149)(72,148)(73,147)(74,146)(75,145)(76,144)(77,143)(78,142)(79,141)(80,168)(81,167)(82,166)(83,165)(84,164)(169,213)(170,212)(171,211)(172,210)(173,209)(174,208)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,201)(182,200)(183,199)(184,198)(185,197)(186,224)(187,223)(188,222)(189,221)(190,220)(191,219)(192,218)(193,217)(194,216)(195,215)(196,214) );
G=PermutationGroup([[(1,210,100,167,132,194,42,75),(2,211,101,168,133,195,43,76),(3,212,102,141,134,196,44,77),(4,213,103,142,135,169,45,78),(5,214,104,143,136,170,46,79),(6,215,105,144,137,171,47,80),(7,216,106,145,138,172,48,81),(8,217,107,146,139,173,49,82),(9,218,108,147,140,174,50,83),(10,219,109,148,113,175,51,84),(11,220,110,149,114,176,52,57),(12,221,111,150,115,177,53,58),(13,222,112,151,116,178,54,59),(14,223,85,152,117,179,55,60),(15,224,86,153,118,180,56,61),(16,197,87,154,119,181,29,62),(17,198,88,155,120,182,30,63),(18,199,89,156,121,183,31,64),(19,200,90,157,122,184,32,65),(20,201,91,158,123,185,33,66),(21,202,92,159,124,186,34,67),(22,203,93,160,125,187,35,68),(23,204,94,161,126,188,36,69),(24,205,95,162,127,189,37,70),(25,206,96,163,128,190,38,71),(26,207,97,164,129,191,39,72),(27,208,98,165,130,192,40,73),(28,209,99,166,131,193,41,74)], [(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)], [(1,138),(2,137),(3,136),(4,135),(5,134),(6,133),(7,132),(8,131),(9,130),(10,129),(11,128),(12,127),(13,126),(14,125),(15,124),(16,123),(17,122),(18,121),(19,120),(20,119),(21,118),(22,117),(23,116),(24,115),(25,114),(26,113),(27,140),(28,139),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,112),(37,111),(38,110),(39,109),(40,108),(41,107),(42,106),(43,105),(44,104),(45,103),(46,102),(47,101),(48,100),(49,99),(50,98),(51,97),(52,96),(53,95),(54,94),(55,93),(56,92),(57,163),(58,162),(59,161),(60,160),(61,159),(62,158),(63,157),(64,156),(65,155),(66,154),(67,153),(68,152),(69,151),(70,150),(71,149),(72,148),(73,147),(74,146),(75,145),(76,144),(77,143),(78,142),(79,141),(80,168),(81,167),(82,166),(83,165),(84,164),(169,213),(170,212),(171,211),(172,210),(173,209),(174,208),(175,207),(176,206),(177,205),(178,204),(179,203),(180,202),(181,201),(182,200),(183,199),(184,198),(185,197),(186,224),(187,223),(188,222),(189,221),(190,220),(191,219),(192,218),(193,217),(194,216),(195,215),(196,214)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D7 | C4○D4 | D14 | D14 | C8○D4 | D28 | C4×D7 | C8×D7 | C4○D28 | D28.2C4 |
| kernel | C8×D28 | C28⋊C8 | D14⋊C8 | C4×C56 | C4×D28 | D7×C2×C8 | C4⋊Dic7 | D14⋊C4 | C2×D28 | D28 | C56 | C4×C8 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 16 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 12 | 24 | 12 | 24 |
Matrix representation of C8×D28 ►in GL3(𝔽113) generated by
| 69 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 78 | 81 |
| 0 | 64 | 94 |
| 1 | 0 | 0 |
| 0 | 43 | 104 |
| 0 | 17 | 70 |
G:=sub<GL(3,GF(113))| [69,0,0,0,18,0,0,0,18],[1,0,0,0,78,64,0,81,94],[1,0,0,0,43,17,0,104,70] >;
C8×D28 in GAP, Magma, Sage, TeX
C_8\times D_{28} % in TeX
G:=Group("C8xD28"); // GroupNames label
G:=SmallGroup(448,220);
// by ID
G=gap.SmallGroup(448,220);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations