metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D28, C56⋊10D4, C7⋊3(C8⋊D4), C2.D8⋊11D7, C4⋊C4.51D14, C4.54(C2×D28), (C2×C8).66D14, D14⋊2Q8⋊7C2, C28.134(C2×D4), C4⋊D28.9C2, C14.D8⋊23C2, C28.41(C4○D4), C14.Q16⋊20C2, (C2×Dic7).50D4, (C22×D7).29D4, C22.232(D4×D7), C2.24(D8⋊D7), C2.20(C4⋊D28), C14.47(C4⋊D4), C14.43(C8⋊C22), (C2×C56).144C22, (C2×C28).302C23, C4.10(Q8⋊2D7), (C2×D28).84C22, C2.23(Q16⋊D7), C14.71(C8.C22), (C2×Dic14).90C22, (C7×C2.D8)⋊8C2, (C2×C8⋊D7)⋊6C2, (C2×C56⋊C2)⋊22C2, (C2×C7⋊C8).72C22, (C2×C4×D7).41C22, (C2×C14).307(C2×D4), (C7×C4⋊C4).95C22, (C2×C4).405(C22×D7), SmallGroup(448,420)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊3D28
G = < a,b,c | a8=b28=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >
Subgroups: 780 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C8⋊D4, C8⋊D7, C56⋊C2, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C2×D28, C14.D8, C14.Q16, C7×C2.D8, C4⋊D28, D14⋊2Q8, C2×C8⋊D7, C2×C56⋊C2, C8⋊3D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C8.C22, D28, C22×D7, C8⋊D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, D8⋊D7, Q16⋊D7, C8⋊3D28
►On 224 points
Generators in S224
(1 218 126 81 158 30 96 190)(2 191 97 31 159 82 127 219)(3 220 128 83 160 32 98 192)(4 193 99 33 161 84 129 221)(5 222 130 57 162 34 100 194)(6 195 101 35 163 58 131 223)(7 224 132 59 164 36 102 196)(8 169 103 37 165 60 133 197)(9 198 134 61 166 38 104 170)(10 171 105 39 167 62 135 199)(11 200 136 63 168 40 106 172)(12 173 107 41 141 64 137 201)(13 202 138 65 142 42 108 174)(14 175 109 43 143 66 139 203)(15 204 140 67 144 44 110 176)(16 177 111 45 145 68 113 205)(17 206 114 69 146 46 112 178)(18 179 85 47 147 70 115 207)(19 208 116 71 148 48 86 180)(20 181 87 49 149 72 117 209)(21 210 118 73 150 50 88 182)(22 183 89 51 151 74 119 211)(23 212 120 75 152 52 90 184)(24 185 91 53 153 76 121 213)(25 214 122 77 154 54 92 186)(26 187 93 55 155 78 123 215)(27 216 124 79 156 56 94 188)(28 189 95 29 157 80 125 217)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 218)(30 217)(31 216)(32 215)(33 214)(34 213)(35 212)(36 211)(37 210)(38 209)(39 208)(40 207)(41 206)(42 205)(43 204)(44 203)(45 202)(46 201)(47 200)(48 199)(49 198)(50 197)(51 224)(52 223)(53 222)(54 221)(55 220)(56 219)(57 185)(58 184)(59 183)(60 182)(61 181)(62 180)(63 179)(64 178)(65 177)(66 176)(67 175)(68 174)(69 173)(70 172)(71 171)(72 170)(73 169)(74 196)(75 195)(76 194)(77 193)(78 192)(79 191)(80 190)(81 189)(82 188)(83 187)(84 186)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(107 112)(108 111)(109 110)(113 138)(114 137)(115 136)(116 135)(117 134)(118 133)(119 132)(120 131)(121 130)(122 129)(123 128)(124 127)(125 126)(139 140)(141 146)(142 145)(143 144)(147 168)(148 167)(149 166)(150 165)(151 164)(152 163)(153 162)(154 161)(155 160)(156 159)(157 158)
G:=sub<Sym(224)| (1,218,126,81,158,30,96,190)(2,191,97,31,159,82,127,219)(3,220,128,83,160,32,98,192)(4,193,99,33,161,84,129,221)(5,222,130,57,162,34,100,194)(6,195,101,35,163,58,131,223)(7,224,132,59,164,36,102,196)(8,169,103,37,165,60,133,197)(9,198,134,61,166,38,104,170)(10,171,105,39,167,62,135,199)(11,200,136,63,168,40,106,172)(12,173,107,41,141,64,137,201)(13,202,138,65,142,42,108,174)(14,175,109,43,143,66,139,203)(15,204,140,67,144,44,110,176)(16,177,111,45,145,68,113,205)(17,206,114,69,146,46,112,178)(18,179,85,47,147,70,115,207)(19,208,116,71,148,48,86,180)(20,181,87,49,149,72,117,209)(21,210,118,73,150,50,88,182)(22,183,89,51,151,74,119,211)(23,212,120,75,152,52,90,184)(24,185,91,53,153,76,121,213)(25,214,122,77,154,54,92,186)(26,187,93,55,155,78,123,215)(27,216,124,79,156,56,94,188)(28,189,95,29,157,80,125,217), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,218)(30,217)(31,216)(32,215)(33,214)(34,213)(35,212)(36,211)(37,210)(38,209)(39,208)(40,207)(41,206)(42,205)(43,204)(44,203)(45,202)(46,201)(47,200)(48,199)(49,198)(50,197)(51,224)(52,223)(53,222)(54,221)(55,220)(56,219)(57,185)(58,184)(59,183)(60,182)(61,181)(62,180)(63,179)(64,178)(65,177)(66,176)(67,175)(68,174)(69,173)(70,172)(71,171)(72,170)(73,169)(74,196)(75,195)(76,194)(77,193)(78,192)(79,191)(80,190)(81,189)(82,188)(83,187)(84,186)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)(139,140)(141,146)(142,145)(143,144)(147,168)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(157,158)>;
G:=Group( (1,218,126,81,158,30,96,190)(2,191,97,31,159,82,127,219)(3,220,128,83,160,32,98,192)(4,193,99,33,161,84,129,221)(5,222,130,57,162,34,100,194)(6,195,101,35,163,58,131,223)(7,224,132,59,164,36,102,196)(8,169,103,37,165,60,133,197)(9,198,134,61,166,38,104,170)(10,171,105,39,167,62,135,199)(11,200,136,63,168,40,106,172)(12,173,107,41,141,64,137,201)(13,202,138,65,142,42,108,174)(14,175,109,43,143,66,139,203)(15,204,140,67,144,44,110,176)(16,177,111,45,145,68,113,205)(17,206,114,69,146,46,112,178)(18,179,85,47,147,70,115,207)(19,208,116,71,148,48,86,180)(20,181,87,49,149,72,117,209)(21,210,118,73,150,50,88,182)(22,183,89,51,151,74,119,211)(23,212,120,75,152,52,90,184)(24,185,91,53,153,76,121,213)(25,214,122,77,154,54,92,186)(26,187,93,55,155,78,123,215)(27,216,124,79,156,56,94,188)(28,189,95,29,157,80,125,217), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,218)(30,217)(31,216)(32,215)(33,214)(34,213)(35,212)(36,211)(37,210)(38,209)(39,208)(40,207)(41,206)(42,205)(43,204)(44,203)(45,202)(46,201)(47,200)(48,199)(49,198)(50,197)(51,224)(52,223)(53,222)(54,221)(55,220)(56,219)(57,185)(58,184)(59,183)(60,182)(61,181)(62,180)(63,179)(64,178)(65,177)(66,176)(67,175)(68,174)(69,173)(70,172)(71,171)(72,170)(73,169)(74,196)(75,195)(76,194)(77,193)(78,192)(79,191)(80,190)(81,189)(82,188)(83,187)(84,186)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)(139,140)(141,146)(142,145)(143,144)(147,168)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(157,158) );
G=PermutationGroup([[(1,218,126,81,158,30,96,190),(2,191,97,31,159,82,127,219),(3,220,128,83,160,32,98,192),(4,193,99,33,161,84,129,221),(5,222,130,57,162,34,100,194),(6,195,101,35,163,58,131,223),(7,224,132,59,164,36,102,196),(8,169,103,37,165,60,133,197),(9,198,134,61,166,38,104,170),(10,171,105,39,167,62,135,199),(11,200,136,63,168,40,106,172),(12,173,107,41,141,64,137,201),(13,202,138,65,142,42,108,174),(14,175,109,43,143,66,139,203),(15,204,140,67,144,44,110,176),(16,177,111,45,145,68,113,205),(17,206,114,69,146,46,112,178),(18,179,85,47,147,70,115,207),(19,208,116,71,148,48,86,180),(20,181,87,49,149,72,117,209),(21,210,118,73,150,50,88,182),(22,183,89,51,151,74,119,211),(23,212,120,75,152,52,90,184),(24,185,91,53,153,76,121,213),(25,214,122,77,154,54,92,186),(26,187,93,55,155,78,123,215),(27,216,124,79,156,56,94,188),(28,189,95,29,157,80,125,217)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,218),(30,217),(31,216),(32,215),(33,214),(34,213),(35,212),(36,211),(37,210),(38,209),(39,208),(40,207),(41,206),(42,205),(43,204),(44,203),(45,202),(46,201),(47,200),(48,199),(49,198),(50,197),(51,224),(52,223),(53,222),(54,221),(55,220),(56,219),(57,185),(58,184),(59,183),(60,182),(61,181),(62,180),(63,179),(64,178),(65,177),(66,176),(67,175),(68,174),(69,173),(70,172),(71,171),(72,170),(73,169),(74,196),(75,195),(76,194),(77,193),(78,192),(79,191),(80,190),(81,189),(82,188),(83,187),(84,186),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(107,112),(108,111),(109,110),(113,138),(114,137),(115,136),(116,135),(117,134),(118,133),(119,132),(120,131),(121,130),(122,129),(123,128),(124,127),(125,126),(139,140),(141,146),(142,145),(143,144),(147,168),(148,167),(149,166),(150,165),(151,164),(152,163),(153,162),(154,161),(155,160),(156,159),(157,158)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 56 | 2 | 2 | 8 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D28 | C8⋊C22 | C8.C22 | Q8⋊2D7 | D4×D7 | D8⋊D7 | Q16⋊D7 |
| kernel | C8⋊3D28 | C14.D8 | C14.Q16 | C7×C2.D8 | C4⋊D28 | D14⋊2Q8 | C2×C8⋊D7 | C2×C56⋊C2 | C56 | C2×Dic7 | C22×D7 | C2.D8 | C28 | C4⋊C4 | C2×C8 | C8 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C8⋊3D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 105 | 66 | 8 | 47 |
| 0 | 0 | 47 | 8 | 66 | 105 |
| 0 | 0 | 109 | 33 | 0 | 0 |
| 0 | 0 | 80 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 79 | 50 | 68 |
| 0 | 0 | 34 | 1 | 45 | 111 |
| 0 | 0 | 0 | 0 | 25 | 34 |
| 0 | 0 | 0 | 0 | 79 | 112 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 79 | 50 | 68 |
| 0 | 0 | 25 | 25 | 63 | 63 |
| 0 | 0 | 0 | 0 | 25 | 34 |
| 0 | 0 | 0 | 0 | 88 | 88 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,105,47,109,80,0,0,66,8,33,4,0,0,8,66,0,0,0,0,47,105,0,0],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,88,34,0,0,0,0,79,1,0,0,0,0,50,45,25,79,0,0,68,111,34,112],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,88,25,0,0,0,0,79,25,0,0,0,0,50,63,25,88,0,0,68,63,34,88] >;
C8⋊3D28 in GAP, Magma, Sage, TeX
C_8\rtimes_3D_{28} % in TeX
G:=Group("C8:3D28"); // GroupNames label
G:=SmallGroup(448,420);
// by ID
G=gap.SmallGroup(448,420);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,254,219,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations