direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C2×C10, C40⋊13C23, C20.78C24, (C22×C8)⋊7C10, C8⋊2(C22×C10), C4.18(D4×C10), (C22×C40)⋊21C2, (C2×C40)⋊50C22, C20.325(C2×D4), (C2×C20).432D4, D4⋊1(C22×C10), (C5×D4)⋊12C23, C4.1(C23×C10), C23.60(C5×D4), (C22×D4)⋊10C10, (D4×C10)⋊65C22, C22.65(D4×C10), (C2×C20).971C23, (C22×C10).221D4, C10.199(C22×D4), (C22×C20).601C22, (D4×C2×C10)⋊25C2, C2.23(D4×C2×C10), (C2×C8)⋊12(C2×C10), (C2×C4).88(C5×D4), (C2×D4)⋊14(C2×C10), (C2×C10).686(C2×D4), (C22×C4).128(C2×C10), (C2×C4).141(C22×C10), SmallGroup(320,1571)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C2×C10
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 658 in 338 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, D4, D4, C23, C23, C10, C10, C10, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C22×C8, C2×D8, C22×D4, C40, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C22×D8, C2×C40, C5×D8, C22×C20, D4×C10, D4×C10, C23×C10, C22×C40, C10×D8, D4×C2×C10, D8×C2×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C24, C2×C10, C2×D8, C22×D4, C5×D4, C22×C10, C22×D8, C5×D8, D4×C10, C23×C10, C10×D8, D4×C2×C10, D8×C2×C10
►On 160 points
Generators in S160
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 118)(12 119)(13 120)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 149)(22 150)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 121)(40 122)(41 110)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 71)(58 72)(59 73)(60 74)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 81)(131 155)(132 156)(133 157)(134 158)(135 159)(136 160)(137 151)(138 152)(139 153)(140 154)
(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)
(1 22 65 16 59 34 41 158)(2 23 66 17 60 35 42 159)(3 24 67 18 51 36 43 160)(4 25 68 19 52 37 44 151)(5 26 69 20 53 38 45 152)(6 27 70 11 54 39 46 153)(7 28 61 12 55 40 47 154)(8 29 62 13 56 31 48 155)(9 30 63 14 57 32 49 156)(10 21 64 15 58 33 50 157)(71 124 108 132 92 148 84 111)(72 125 109 133 93 149 85 112)(73 126 110 134 94 150 86 113)(74 127 101 135 95 141 87 114)(75 128 102 136 96 142 88 115)(76 129 103 137 97 143 89 116)(77 130 104 138 98 144 90 117)(78 121 105 139 99 145 81 118)(79 122 106 140 100 146 82 119)(80 123 107 131 91 147 83 120)
(1 158)(2 159)(3 160)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 157)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 51)(19 52)(20 53)(21 50)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 61)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)
G:=sub<Sym(160)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,118)(12,119)(13,120)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,149)(22,150)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,121)(40,122)(41,110)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,71)(58,72)(59,73)(60,74)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,81)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)(137,151)(138,152)(139,153)(140,154), (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), (1,22,65,16,59,34,41,158)(2,23,66,17,60,35,42,159)(3,24,67,18,51,36,43,160)(4,25,68,19,52,37,44,151)(5,26,69,20,53,38,45,152)(6,27,70,11,54,39,46,153)(7,28,61,12,55,40,47,154)(8,29,62,13,56,31,48,155)(9,30,63,14,57,32,49,156)(10,21,64,15,58,33,50,157)(71,124,108,132,92,148,84,111)(72,125,109,133,93,149,85,112)(73,126,110,134,94,150,86,113)(74,127,101,135,95,141,87,114)(75,128,102,136,96,142,88,115)(76,129,103,137,97,143,89,116)(77,130,104,138,98,144,90,117)(78,121,105,139,99,145,81,118)(79,122,106,140,100,146,82,119)(80,123,107,131,91,147,83,120), (1,158)(2,159)(3,160)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,51)(19,52)(20,53)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,61)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)>;
G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,91)(9,92)(10,93)(11,118)(12,119)(13,120)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,149)(22,150)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,121)(40,122)(41,110)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,71)(58,72)(59,73)(60,74)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,81)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)(137,151)(138,152)(139,153)(140,154), (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), (1,22,65,16,59,34,41,158)(2,23,66,17,60,35,42,159)(3,24,67,18,51,36,43,160)(4,25,68,19,52,37,44,151)(5,26,69,20,53,38,45,152)(6,27,70,11,54,39,46,153)(7,28,61,12,55,40,47,154)(8,29,62,13,56,31,48,155)(9,30,63,14,57,32,49,156)(10,21,64,15,58,33,50,157)(71,124,108,132,92,148,84,111)(72,125,109,133,93,149,85,112)(73,126,110,134,94,150,86,113)(74,127,101,135,95,141,87,114)(75,128,102,136,96,142,88,115)(76,129,103,137,97,143,89,116)(77,130,104,138,98,144,90,117)(78,121,105,139,99,145,81,118)(79,122,106,140,100,146,82,119)(80,123,107,131,91,147,83,120), (1,158)(2,159)(3,160)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,51)(19,52)(20,53)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,61)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150) );
G=PermutationGroup([[(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,91),(9,92),(10,93),(11,118),(12,119),(13,120),(14,111),(15,112),(16,113),(17,114),(18,115),(19,116),(20,117),(21,149),(22,150),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,121),(40,122),(41,110),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,109),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,71),(58,72),(59,73),(60,74),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,81),(131,155),(132,156),(133,157),(134,158),(135,159),(136,160),(137,151),(138,152),(139,153),(140,154)], [(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)], [(1,22,65,16,59,34,41,158),(2,23,66,17,60,35,42,159),(3,24,67,18,51,36,43,160),(4,25,68,19,52,37,44,151),(5,26,69,20,53,38,45,152),(6,27,70,11,54,39,46,153),(7,28,61,12,55,40,47,154),(8,29,62,13,56,31,48,155),(9,30,63,14,57,32,49,156),(10,21,64,15,58,33,50,157),(71,124,108,132,92,148,84,111),(72,125,109,133,93,149,85,112),(73,126,110,134,94,150,86,113),(74,127,101,135,95,141,87,114),(75,128,102,136,96,142,88,115),(76,129,103,137,97,143,89,116),(77,130,104,138,98,144,90,117),(78,121,105,139,99,145,81,118),(79,122,106,140,100,146,82,119),(80,123,107,131,91,147,83,120)], [(1,158),(2,159),(3,160),(4,151),(5,152),(6,153),(7,154),(8,155),(9,156),(10,157),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,51),(19,52),(20,53),(21,50),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,61),(71,111),(72,112),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 10AC | ··· | 10BH | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | D8 | C5×D4 | C5×D4 | C5×D8 |
| kernel | D8×C2×C10 | C22×C40 | C10×D8 | D4×C2×C10 | C22×D8 | C22×C8 | C2×D8 | C22×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 2 | 4 | 4 | 48 | 8 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of D8×C2×C10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 29 | 29 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,0,0,0,0,0,0,23,0,0,0,0,0,0,25,0,0,0,0,0,0,25,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,29,0,0,0,0,29,29] >;
D8×C2×C10 in GAP, Magma, Sage, TeX
D_8\times C_2\times C_{10} % in TeX
G:=Group("D8xC2xC10"); // GroupNames label
G:=SmallGroup(320,1571);
// by ID
G=gap.SmallGroup(320,1571);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations