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, (C22×C10).23(C2×C4), (C2×C10).75(C22×C4), (C2×Dic5).70(C2×C4), SmallGroup(320,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C5⋊C8
G = < a,b,c,d | a4=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 394 in 134 conjugacy classes, 64 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C8×D4, C4×Dic5, C4⋊Dic5, C23.D5, C2×C5⋊C8, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5, D4×Dic5, C22×C5⋊C8, D4×C5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C5⋊C8, C2×F5, C8×D4, C2×C5⋊C8, C22×F5, D4.F5, D4×F5, C22×C5⋊C8, D4×C5⋊C8
►On 160 points
Generators in S160
(1 95 132 18)(2 96 133 19)(3 89 134 20)(4 90 135 21)(5 91 136 22)(6 92 129 23)(7 93 130 24)(8 94 131 17)(9 126 80 101)(10 127 73 102)(11 128 74 103)(12 121 75 104)(13 122 76 97)(14 123 77 98)(15 124 78 99)(16 125 79 100)(25 47 33 139)(26 48 34 140)(27 41 35 141)(28 42 36 142)(29 43 37 143)(30 44 38 144)(31 45 39 137)(32 46 40 138)(49 160 106 114)(50 153 107 115)(51 154 108 116)(52 155 109 117)(53 156 110 118)(54 157 111 119)(55 158 112 120)(56 159 105 113)(57 83 65 151)(58 84 66 152)(59 85 67 145)(60 86 68 146)(61 87 69 147)(62 88 70 148)(63 81 71 149)(64 82 72 150)
(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 105)(81 149)(82 150)(83 151)(84 152)(85 145)(86 146)(87 147)(88 148)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 121)
(1 12 57 143 113)(2 144 13 114 58)(3 115 137 59 14)(4 60 116 15 138)(5 16 61 139 117)(6 140 9 118 62)(7 119 141 63 10)(8 64 120 11 142)(17 150 112 103 36)(18 104 151 37 105)(19 38 97 106 152)(20 107 39 145 98)(21 146 108 99 40)(22 100 147 33 109)(23 34 101 110 148)(24 111 35 149 102)(25 52 91 125 87)(26 126 53 88 92)(27 81 127 93 54)(28 94 82 55 128)(29 56 95 121 83)(30 122 49 84 96)(31 85 123 89 50)(32 90 86 51 124)(41 71 73 130 157)(42 131 72 158 74)(43 159 132 75 65)(44 76 160 66 133)(45 67 77 134 153)(46 135 68 154 78)(47 155 136 79 69)(48 80 156 70 129)
(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)| (1,95,132,18)(2,96,133,19)(3,89,134,20)(4,90,135,21)(5,91,136,22)(6,92,129,23)(7,93,130,24)(8,94,131,17)(9,126,80,101)(10,127,73,102)(11,128,74,103)(12,121,75,104)(13,122,76,97)(14,123,77,98)(15,124,78,99)(16,125,79,100)(25,47,33,139)(26,48,34,140)(27,41,35,141)(28,42,36,142)(29,43,37,143)(30,44,38,144)(31,45,39,137)(32,46,40,138)(49,160,106,114)(50,153,107,115)(51,154,108,116)(52,155,109,117)(53,156,110,118)(54,157,111,119)(55,158,112,120)(56,159,105,113)(57,83,65,151)(58,84,66,152)(59,85,67,145)(60,86,68,146)(61,87,69,147)(62,88,70,148)(63,81,71,149)(64,82,72,150), (17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,105)(81,149)(82,150)(83,151)(84,152)(85,145)(86,146)(87,147)(88,148)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,121), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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( (1,95,132,18)(2,96,133,19)(3,89,134,20)(4,90,135,21)(5,91,136,22)(6,92,129,23)(7,93,130,24)(8,94,131,17)(9,126,80,101)(10,127,73,102)(11,128,74,103)(12,121,75,104)(13,122,76,97)(14,123,77,98)(15,124,78,99)(16,125,79,100)(25,47,33,139)(26,48,34,140)(27,41,35,141)(28,42,36,142)(29,43,37,143)(30,44,38,144)(31,45,39,137)(32,46,40,138)(49,160,106,114)(50,153,107,115)(51,154,108,116)(52,155,109,117)(53,156,110,118)(54,157,111,119)(55,158,112,120)(56,159,105,113)(57,83,65,151)(58,84,66,152)(59,85,67,145)(60,86,68,146)(61,87,69,147)(62,88,70,148)(63,81,71,149)(64,82,72,150), (17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,105)(81,149)(82,150)(83,151)(84,152)(85,145)(86,146)(87,147)(88,148)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,121), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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([[(1,95,132,18),(2,96,133,19),(3,89,134,20),(4,90,135,21),(5,91,136,22),(6,92,129,23),(7,93,130,24),(8,94,131,17),(9,126,80,101),(10,127,73,102),(11,128,74,103),(12,121,75,104),(13,122,76,97),(14,123,77,98),(15,124,78,99),(16,125,79,100),(25,47,33,139),(26,48,34,140),(27,41,35,141),(28,42,36,142),(29,43,37,143),(30,44,38,144),(31,45,39,137),(32,46,40,138),(49,160,106,114),(50,153,107,115),(51,154,108,116),(52,155,109,117),(53,156,110,118),(54,157,111,119),(55,158,112,120),(56,159,105,113),(57,83,65,151),(58,84,66,152),(59,85,67,145),(60,86,68,146),(61,87,69,147),(62,88,70,148),(63,81,71,149),(64,82,72,150)], [(17,94),(18,95),(19,96),(20,89),(21,90),(22,91),(23,92),(24,93),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,105),(81,149),(82,150),(83,151),(84,152),(85,145),(86,146),(87,147),(88,148),(97,122),(98,123),(99,124),(100,125),(101,126),(102,127),(103,128),(104,121)], [(1,12,57,143,113),(2,144,13,114,58),(3,115,137,59,14),(4,60,116,15,138),(5,16,61,139,117),(6,140,9,118,62),(7,119,141,63,10),(8,64,120,11,142),(17,150,112,103,36),(18,104,151,37,105),(19,38,97,106,152),(20,107,39,145,98),(21,146,108,99,40),(22,100,147,33,109),(23,34,101,110,148),(24,111,35,149,102),(25,52,91,125,87),(26,126,53,88,92),(27,81,127,93,54),(28,94,82,55,128),(29,56,95,121,83),(30,122,49,84,96),(31,85,123,89,50),(32,90,86,51,124),(41,71,73,130,157),(42,131,72,158,74),(43,159,132,75,65),(44,76,160,66,133),(45,67,77,134,153),(46,135,68,154,78),(47,155,136,79,69),(48,80,156,70,129)], [(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)]])
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 |
Matrix representation of D4×C5⋊C8 ►in 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] >;
D4×C5⋊C8 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