metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.49(C2×D8), (C2×C10).40D8, C4⋊C4.227D10, C20⋊7D4.9C2, (C2×C20).283D4, D20⋊6C4⋊25C2, C4.86(C4○D20), C10.D8⋊25C2, C20.55D4⋊4C2, C22.9(D4⋊D5), (C22×C4).94D10, C5⋊4(C22.D8), C20.174(C4○D4), (C2×C20).320C23, (C2×D20).94C22, (C22×C10).185D4, C23.77(C5⋊D4), C2.6(C20.C23), C10.84(C8.C22), C4⋊Dic5.130C22, (C22×C20).135C22, C2.8(C23.23D10), C10.58(C22.D4), (C2×C4⋊C4)⋊3D5, (C10×C4⋊C4)⋊3C2, C2.5(C2×D4⋊D5), (C2×C10).440(C2×D4), (C2×C4).31(C5⋊D4), (C5×C4⋊C4).258C22, (C2×C5⋊2C8).81C22, (C2×C4).420(C22×D5), C22.130(C2×C5⋊D4), SmallGroup(320,594)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.(C2×D8)
G = < a,b,c,d | a10=b2=c8=1, d2=a5, ab=ba, cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >
Subgroups: 462 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C5⋊2C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.D8, C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C10.D8 [×2], D20⋊6C4 [×2], C20.55D4, C20⋊7D4, C10×C4⋊C4, C10.(C2×D8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, C5⋊D4 [×2], C22×D5, C22.D8, D4⋊D5 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, C2×D4⋊D5, C20.C23, C10.(C2×D8)
►On 160 points
Generators in S160
(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)
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 71)(18 72)(19 73)(20 74)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)(89 159)(90 160)
(1 75 45 70 30 85 35 60)(2 74 46 69 21 84 36 59)(3 73 47 68 22 83 37 58)(4 72 48 67 23 82 38 57)(5 71 49 66 24 81 39 56)(6 80 50 65 25 90 40 55)(7 79 41 64 26 89 31 54)(8 78 42 63 27 88 32 53)(9 77 43 62 28 87 33 52)(10 76 44 61 29 86 34 51)(11 120 140 95 155 130 150 105)(12 119 131 94 156 129 141 104)(13 118 132 93 157 128 142 103)(14 117 133 92 158 127 143 102)(15 116 134 91 159 126 144 101)(16 115 135 100 160 125 145 110)(17 114 136 99 151 124 146 109)(18 113 137 98 152 123 147 108)(19 112 138 97 153 122 148 107)(20 111 139 96 154 121 149 106)
(1 135 6 140)(2 136 7 131)(3 137 8 132)(4 138 9 133)(5 139 10 134)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(51 91 56 96)(52 92 57 97)(53 93 58 98)(54 94 59 99)(55 95 60 100)(61 101 66 106)(62 102 67 107)(63 103 68 108)(64 104 69 109)(65 105 70 110)(71 111 76 116)(72 112 77 117)(73 113 78 118)(74 114 79 119)(75 115 80 120)(81 121 86 126)(82 122 87 127)(83 123 88 128)(84 124 89 129)(85 125 90 130)
G:=sub<Sym(160)| (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), (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,71)(18,72)(19,73)(20,74)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,159)(90,160), (1,75,45,70,30,85,35,60)(2,74,46,69,21,84,36,59)(3,73,47,68,22,83,37,58)(4,72,48,67,23,82,38,57)(5,71,49,66,24,81,39,56)(6,80,50,65,25,90,40,55)(7,79,41,64,26,89,31,54)(8,78,42,63,27,88,32,53)(9,77,43,62,28,87,33,52)(10,76,44,61,29,86,34,51)(11,120,140,95,155,130,150,105)(12,119,131,94,156,129,141,104)(13,118,132,93,157,128,142,103)(14,117,133,92,158,127,143,102)(15,116,134,91,159,126,144,101)(16,115,135,100,160,125,145,110)(17,114,136,99,151,124,146,109)(18,113,137,98,152,123,147,108)(19,112,138,97,153,122,148,107)(20,111,139,96,154,121,149,106), (1,135,6,140)(2,136,7,131)(3,137,8,132)(4,138,9,133)(5,139,10,134)(11,45,16,50)(12,46,17,41)(13,47,18,42)(14,48,19,43)(15,49,20,44)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(25,150,30,145)(31,156,36,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(51,91,56,96)(52,92,57,97)(53,93,58,98)(54,94,59,99)(55,95,60,100)(61,101,66,106)(62,102,67,107)(63,103,68,108)(64,104,69,109)(65,105,70,110)(71,111,76,116)(72,112,77,117)(73,113,78,118)(74,114,79,119)(75,115,80,120)(81,121,86,126)(82,122,87,127)(83,123,88,128)(84,124,89,129)(85,125,90,130)>;
G:=Group( (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), (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,71)(18,72)(19,73)(20,74)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(81,151)(82,152)(83,153)(84,154)(85,155)(86,156)(87,157)(88,158)(89,159)(90,160), (1,75,45,70,30,85,35,60)(2,74,46,69,21,84,36,59)(3,73,47,68,22,83,37,58)(4,72,48,67,23,82,38,57)(5,71,49,66,24,81,39,56)(6,80,50,65,25,90,40,55)(7,79,41,64,26,89,31,54)(8,78,42,63,27,88,32,53)(9,77,43,62,28,87,33,52)(10,76,44,61,29,86,34,51)(11,120,140,95,155,130,150,105)(12,119,131,94,156,129,141,104)(13,118,132,93,157,128,142,103)(14,117,133,92,158,127,143,102)(15,116,134,91,159,126,144,101)(16,115,135,100,160,125,145,110)(17,114,136,99,151,124,146,109)(18,113,137,98,152,123,147,108)(19,112,138,97,153,122,148,107)(20,111,139,96,154,121,149,106), (1,135,6,140)(2,136,7,131)(3,137,8,132)(4,138,9,133)(5,139,10,134)(11,45,16,50)(12,46,17,41)(13,47,18,42)(14,48,19,43)(15,49,20,44)(21,146,26,141)(22,147,27,142)(23,148,28,143)(24,149,29,144)(25,150,30,145)(31,156,36,151)(32,157,37,152)(33,158,38,153)(34,159,39,154)(35,160,40,155)(51,91,56,96)(52,92,57,97)(53,93,58,98)(54,94,59,99)(55,95,60,100)(61,101,66,106)(62,102,67,107)(63,103,68,108)(64,104,69,109)(65,105,70,110)(71,111,76,116)(72,112,77,117)(73,113,78,118)(74,114,79,119)(75,115,80,120)(81,121,86,126)(82,122,87,127)(83,123,88,128)(84,124,89,129)(85,125,90,130) );
G=PermutationGroup([(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)], [(1,110),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,71),(18,72),(19,73),(20,74),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,99),(30,100),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,127),(38,128),(39,129),(40,130),(41,111),(42,112),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,131),(62,132),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(81,151),(82,152),(83,153),(84,154),(85,155),(86,156),(87,157),(88,158),(89,159),(90,160)], [(1,75,45,70,30,85,35,60),(2,74,46,69,21,84,36,59),(3,73,47,68,22,83,37,58),(4,72,48,67,23,82,38,57),(5,71,49,66,24,81,39,56),(6,80,50,65,25,90,40,55),(7,79,41,64,26,89,31,54),(8,78,42,63,27,88,32,53),(9,77,43,62,28,87,33,52),(10,76,44,61,29,86,34,51),(11,120,140,95,155,130,150,105),(12,119,131,94,156,129,141,104),(13,118,132,93,157,128,142,103),(14,117,133,92,158,127,143,102),(15,116,134,91,159,126,144,101),(16,115,135,100,160,125,145,110),(17,114,136,99,151,124,146,109),(18,113,137,98,152,123,147,108),(19,112,138,97,153,122,148,107),(20,111,139,96,154,121,149,106)], [(1,135,6,140),(2,136,7,131),(3,137,8,132),(4,138,9,133),(5,139,10,134),(11,45,16,50),(12,46,17,41),(13,47,18,42),(14,48,19,43),(15,49,20,44),(21,146,26,141),(22,147,27,142),(23,148,28,143),(24,149,29,144),(25,150,30,145),(31,156,36,151),(32,157,37,152),(33,158,38,153),(34,159,39,154),(35,160,40,155),(51,91,56,96),(52,92,57,97),(53,93,58,98),(54,94,59,99),(55,95,60,100),(61,101,66,106),(62,102,67,107),(63,103,68,108),(64,104,69,109),(65,105,70,110),(71,111,76,116),(72,112,77,117),(73,113,78,118),(74,114,79,119),(75,115,80,120),(81,121,86,126),(82,122,87,127),(83,123,88,128),(84,124,89,129),(85,125,90,130)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 4 | ··· | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D8 | D10 | D10 | C5⋊D4 | C5⋊D4 | C4○D20 | C8.C22 | D4⋊D5 | C20.C23 |
| kernel | C10.(C2×D8) | C10.D8 | D20⋊6C4 | C20.55D4 | C20⋊7D4 | C10×C4⋊C4 | C2×C20 | C22×C10 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C23 | C4 | C10 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C10.(C2×D8) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 25 | 1 | 0 | 0 | 0 | 0 |
| 32 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 20 | 9 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 0 | 0 | 0 | 0 | 29 | 12 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,32,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,0,0,12,29,0,0,0,0,12,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,29,29,0,0,0,0,29,12] >;
C10.(C2×D8) in GAP, Magma, Sage, TeX
C_{10}.(C_2\times D_8) % in TeX
G:=Group("C10.(C2xD8)"); // GroupNames label
G:=SmallGroup(320,594);
// by ID
G=gap.SmallGroup(320,594);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations