direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×C5⋊C8, C5⋊3(C8×D4), C20⋊2(C2×C8), (C5×D4)⋊2C8, C2.5(D4×F5), C20⋊C8⋊5C2, (D4×C10).8C4, C10.26(C4×D4), (C2×D4).12F5, C4⋊Dic5.13C4, C2.5(D4.F5), C23.D5.5C4, C23.29(C2×F5), C10.14(C8○D4), C10.21(C22×C8), (D4×Dic5).17C2, Dic5.78(C2×D4), C23.2F5⋊8C2, Dic5.57(C4○D4), C22.51(C22×F5), (C4×Dic5).194C22, (C2×Dic5).351C23, (C22×Dic5).184C22, C4⋊1(C2×C5⋊C8), (C4×C5⋊C8)⋊5C2, C22⋊1(C2×C5⋊C8), (C2×C10)⋊2(C2×C8), (C22×C5⋊C8)⋊4C2, C2.6(C22×C5⋊C8), (C2×C4).80(C2×F5), (C2×C20).54(C2×C4), (C2×C5⋊C8).39C22, (C2×C10).75(C22×C4), (C22×C10).23(C2×C4), (C2×Dic5).70(C2×C4), SmallGroup(320,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 394 in 134 conjugacy classes, 64 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C8 [×5], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×8], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], C8×D4, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×4], C22×Dic5 [×2], D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5 [×2], D4×Dic5, C22×C5⋊C8 [×2], D4×C5⋊C8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C5⋊C8 [×4], C2×F5 [×3], C8×D4, C2×C5⋊C8 [×6], C22×F5, D4.F5, D4×F5, C22×C5⋊C8, D4×C5⋊C8
Generators and relations
G = < a,b,c,d | a4=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►On 160 points
Generators in S160
(1 12 137 18)(2 13 138 19)(3 14 139 20)(4 15 140 21)(5 16 141 22)(6 9 142 23)(7 10 143 24)(8 11 144 17)(25 47 151 54)(26 48 152 55)(27 41 145 56)(28 42 146 49)(29 43 147 50)(30 44 148 51)(31 45 149 52)(32 46 150 53)(33 136 102 69)(34 129 103 70)(35 130 104 71)(36 131 97 72)(37 132 98 65)(38 133 99 66)(39 134 100 67)(40 135 101 68)(57 155 109 117)(58 156 110 118)(59 157 111 119)(60 158 112 120)(61 159 105 113)(62 160 106 114)(63 153 107 115)(64 154 108 116)(73 82 90 127)(74 83 91 128)(75 84 92 121)(76 85 93 122)(77 86 94 123)(78 87 95 124)(79 88 96 125)(80 81 89 126)
(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 151)(26 152)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 102)(34 103)(35 104)(36 97)(37 98)(38 99)(39 100)(40 101)(57 109)(58 110)(59 111)(60 112)(61 105)(62 106)(63 107)(64 108)(81 126)(82 127)(83 128)(84 121)(85 122)(86 123)(87 124)(88 125)
(1 92 132 50 113)(2 51 93 114 133)(3 115 52 134 94)(4 135 116 95 53)(5 96 136 54 117)(6 55 89 118 129)(7 119 56 130 90)(8 131 120 91 49)(9 26 126 58 103)(10 59 27 104 127)(11 97 60 128 28)(12 121 98 29 61)(13 30 122 62 99)(14 63 31 100 123)(15 101 64 124 32)(16 125 102 25 57)(17 36 112 83 146)(18 84 37 147 105)(19 148 85 106 38)(20 107 149 39 86)(21 40 108 87 150)(22 88 33 151 109)(23 152 81 110 34)(24 111 145 35 82)(41 71 73 143 157)(42 144 72 158 74)(43 159 137 75 65)(44 76 160 66 138)(45 67 77 139 153)(46 140 68 154 78)(47 155 141 79 69)(48 80 156 70 142)
(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)
G:=sub<Sym(160)| (1,12,137,18)(2,13,138,19)(3,14,139,20)(4,15,140,21)(5,16,141,22)(6,9,142,23)(7,10,143,24)(8,11,144,17)(25,47,151,54)(26,48,152,55)(27,41,145,56)(28,42,146,49)(29,43,147,50)(30,44,148,51)(31,45,149,52)(32,46,150,53)(33,136,102,69)(34,129,103,70)(35,130,104,71)(36,131,97,72)(37,132,98,65)(38,133,99,66)(39,134,100,67)(40,135,101,68)(57,155,109,117)(58,156,110,118)(59,157,111,119)(60,158,112,120)(61,159,105,113)(62,160,106,114)(63,153,107,115)(64,154,108,116)(73,82,90,127)(74,83,91,128)(75,84,92,121)(76,85,93,122)(77,86,94,123)(78,87,95,124)(79,88,96,125)(80,81,89,126), (9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,151)(26,152)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,102)(34,103)(35,104)(36,97)(37,98)(38,99)(39,100)(40,101)(57,109)(58,110)(59,111)(60,112)(61,105)(62,106)(63,107)(64,108)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125), (1,92,132,50,113)(2,51,93,114,133)(3,115,52,134,94)(4,135,116,95,53)(5,96,136,54,117)(6,55,89,118,129)(7,119,56,130,90)(8,131,120,91,49)(9,26,126,58,103)(10,59,27,104,127)(11,97,60,128,28)(12,121,98,29,61)(13,30,122,62,99)(14,63,31,100,123)(15,101,64,124,32)(16,125,102,25,57)(17,36,112,83,146)(18,84,37,147,105)(19,148,85,106,38)(20,107,149,39,86)(21,40,108,87,150)(22,88,33,151,109)(23,152,81,110,34)(24,111,145,35,82)(41,71,73,143,157)(42,144,72,158,74)(43,159,137,75,65)(44,76,160,66,138)(45,67,77,139,153)(46,140,68,154,78)(47,155,141,79,69)(48,80,156,70,142), (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)>;
G:=Group( (1,12,137,18)(2,13,138,19)(3,14,139,20)(4,15,140,21)(5,16,141,22)(6,9,142,23)(7,10,143,24)(8,11,144,17)(25,47,151,54)(26,48,152,55)(27,41,145,56)(28,42,146,49)(29,43,147,50)(30,44,148,51)(31,45,149,52)(32,46,150,53)(33,136,102,69)(34,129,103,70)(35,130,104,71)(36,131,97,72)(37,132,98,65)(38,133,99,66)(39,134,100,67)(40,135,101,68)(57,155,109,117)(58,156,110,118)(59,157,111,119)(60,158,112,120)(61,159,105,113)(62,160,106,114)(63,153,107,115)(64,154,108,116)(73,82,90,127)(74,83,91,128)(75,84,92,121)(76,85,93,122)(77,86,94,123)(78,87,95,124)(79,88,96,125)(80,81,89,126), (9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,151)(26,152)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,102)(34,103)(35,104)(36,97)(37,98)(38,99)(39,100)(40,101)(57,109)(58,110)(59,111)(60,112)(61,105)(62,106)(63,107)(64,108)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125), (1,92,132,50,113)(2,51,93,114,133)(3,115,52,134,94)(4,135,116,95,53)(5,96,136,54,117)(6,55,89,118,129)(7,119,56,130,90)(8,131,120,91,49)(9,26,126,58,103)(10,59,27,104,127)(11,97,60,128,28)(12,121,98,29,61)(13,30,122,62,99)(14,63,31,100,123)(15,101,64,124,32)(16,125,102,25,57)(17,36,112,83,146)(18,84,37,147,105)(19,148,85,106,38)(20,107,149,39,86)(21,40,108,87,150)(22,88,33,151,109)(23,152,81,110,34)(24,111,145,35,82)(41,71,73,143,157)(42,144,72,158,74)(43,159,137,75,65)(44,76,160,66,138)(45,67,77,139,153)(46,140,68,154,78)(47,155,141,79,69)(48,80,156,70,142), (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) );
G=PermutationGroup([(1,12,137,18),(2,13,138,19),(3,14,139,20),(4,15,140,21),(5,16,141,22),(6,9,142,23),(7,10,143,24),(8,11,144,17),(25,47,151,54),(26,48,152,55),(27,41,145,56),(28,42,146,49),(29,43,147,50),(30,44,148,51),(31,45,149,52),(32,46,150,53),(33,136,102,69),(34,129,103,70),(35,130,104,71),(36,131,97,72),(37,132,98,65),(38,133,99,66),(39,134,100,67),(40,135,101,68),(57,155,109,117),(58,156,110,118),(59,157,111,119),(60,158,112,120),(61,159,105,113),(62,160,106,114),(63,153,107,115),(64,154,108,116),(73,82,90,127),(74,83,91,128),(75,84,92,121),(76,85,93,122),(77,86,94,123),(78,87,95,124),(79,88,96,125),(80,81,89,126)], [(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22),(25,151),(26,152),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,102),(34,103),(35,104),(36,97),(37,98),(38,99),(39,100),(40,101),(57,109),(58,110),(59,111),(60,112),(61,105),(62,106),(63,107),(64,108),(81,126),(82,127),(83,128),(84,121),(85,122),(86,123),(87,124),(88,125)], [(1,92,132,50,113),(2,51,93,114,133),(3,115,52,134,94),(4,135,116,95,53),(5,96,136,54,117),(6,55,89,118,129),(7,119,56,130,90),(8,131,120,91,49),(9,26,126,58,103),(10,59,27,104,127),(11,97,60,128,28),(12,121,98,29,61),(13,30,122,62,99),(14,63,31,100,123),(15,101,64,124,32),(16,125,102,25,57),(17,36,112,83,146),(18,84,37,147,105),(19,148,85,106,38),(20,107,149,39,86),(21,40,108,87,150),(22,88,33,151,109),(23,152,81,110,34),(24,111,145,35,82),(41,71,73,143,157),(42,144,72,158,74),(43,159,137,75,65),(44,76,160,66,138),(45,67,77,139,153),(46,140,68,154,78),(47,155,141,79,69),(48,80,156,70,142)], [(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)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 11 | 6 | 28 |
| 0 | 0 | 0 | 0 | 21 | 39 | 28 | 2 |
| 0 | 0 | 0 | 0 | 2 | 13 | 39 | 8 |
| 0 | 0 | 0 | 0 | 13 | 19 | 26 | 30 |
G:=sub<GL(8,GF(41))| [40,40,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,21,2,13,0,0,0,0,11,39,13,19,0,0,0,0,6,28,39,26,0,0,0,0,28,2,8,30] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | F5 | C2×F5 | C5⋊C8 | C2×F5 | D4.F5 | D4×F5 |
| kernel | D4×C5⋊C8 | C4×C5⋊C8 | C20⋊C8 | C23.2F5 | D4×Dic5 | C22×C5⋊C8 | C4⋊Dic5 | C23.D5 | D4×C10 | C5×D4 | C5⋊C8 | Dic5 | C10 | C2×D4 | C2×C4 | D4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 16 | 2 | 2 | 4 | 1 | 1 | 4 | 2 | 1 | 1 |
In GAP, Magma, Sage, TeX
D_4\times C_5\rtimes C_8
% in TeX
G:=Group("D4xC5:C8"); // GroupNames label
G:=SmallGroup(320,1110);
// by ID
G=gap.SmallGroup(320,1110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀