direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q16⋊D5, Q16⋊8D10, C40.41C23, C20.11C24, D20.6C23, Dic10.7C23, C4.47(D4×D5), (C2×Q16)⋊11D5, (C4×D5).17D4, C20.86(C2×D4), Q8⋊D5⋊9C22, (Q8×D5)⋊7C22, (C10×Q16)⋊11C2, D10.86(C2×D4), (C2×C8).104D10, C5⋊2C8.4C23, (C4×D5).6C23, C5⋊Q16⋊9C22, C8.13(C22×D5), C4.11(C23×D5), Q8.5(C22×D5), (C5×Q8).5C23, C8⋊D5⋊14C22, C40⋊C2⋊15C22, C10⋊3(C8.C22), (C2×Q8).153D10, Dic5.97(C2×D4), (C5×Q16)⋊12C22, C22.143(D4×D5), (C2×C40).152C22, (C2×C20).528C23, (C2×Dic5).250D4, (C22×D5).137D4, C10.112(C22×D4), Q8⋊2D5.8C22, (C2×D20).185C22, (Q8×C10).150C22, (C2×Dic10).205C22, (C2×Q8×D5)⋊16C2, C2.85(C2×D4×D5), C5⋊3(C2×C8.C22), (C2×C8⋊D5)⋊9C2, (C2×Q8⋊D5)⋊27C2, (C2×C40⋊C2)⋊25C2, (C2×C5⋊Q16)⋊28C2, (C2×C10).401(C2×D4), (C2×Q8⋊2D5).8C2, (C2×C4×D5).167C22, (C2×C4).616(C22×D5), (C2×C5⋊2C8).181C22, SmallGroup(320,1436)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4 [×7], Q8 [×4], Q8 [×9], C23 [×2], D5 [×4], C10, C10 [×2], C2×C8, C2×C8, M4(2) [×4], SD16 [×8], Q16 [×4], Q16 [×4], C22×C4 [×3], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×8], C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×6], C2×C10, C2×M4(2), C2×SD16 [×2], C2×Q16, C2×Q16, C8.C22 [×8], C22×Q8, C2×C4○D4, C5⋊2C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×4], C4×D5 [×8], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C22×D5, C22×D5, C2×C8.C22, C8⋊D5 [×4], C40⋊C2 [×4], C2×C5⋊2C8, Q8⋊D5 [×4], C5⋊Q16 [×4], C2×C40, C5×Q16 [×4], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20, Q8×D5 [×4], Q8×D5 [×2], Q8⋊2D5 [×4], Q8⋊2D5 [×2], Q8×C10 [×2], C2×C8⋊D5, C2×C40⋊C2, Q16⋊D5 [×8], C2×Q8⋊D5, C2×C5⋊Q16, C10×Q16, C2×Q8×D5, C2×Q8⋊2D5, C2×Q16⋊D5
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C22×D5 [×7], C2×C8.C22, D4×D5 [×2], C23×D5, Q16⋊D5 [×2], C2×D4×D5, C2×Q16⋊D5
Generators and relations
G = < a,b,c,d,e | a2=b8=d5=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >
►On 160 points
Generators in S160
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 97)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 157)(18 158)(19 159)(20 160)(21 153)(22 154)(23 155)(24 156)(25 149)(26 150)(27 151)(28 152)(29 145)(30 146)(31 147)(32 148)(33 142)(34 143)(35 144)(36 137)(37 138)(38 139)(39 140)(40 141)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 89)(56 90)(57 132)(58 133)(59 134)(60 135)(61 136)(62 129)(63 130)(64 131)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 126)(74 127)(75 128)(76 121)(77 122)(78 123)(79 124)(80 125)
(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 67 5 71)(2 66 6 70)(3 65 7 69)(4 72 8 68)(9 28 13 32)(10 27 14 31)(11 26 15 30)(12 25 16 29)(17 140 21 144)(18 139 22 143)(19 138 23 142)(20 137 24 141)(33 159 37 155)(34 158 38 154)(35 157 39 153)(36 156 40 160)(41 75 45 79)(42 74 46 78)(43 73 47 77)(44 80 48 76)(49 59 53 63)(50 58 54 62)(51 57 55 61)(52 64 56 60)(81 124 85 128)(82 123 86 127)(83 122 87 126)(84 121 88 125)(89 136 93 132)(90 135 94 131)(91 134 95 130)(92 133 96 129)(97 118 101 114)(98 117 102 113)(99 116 103 120)(100 115 104 119)(105 146 109 150)(106 145 110 149)(107 152 111 148)(108 151 112 147)
(1 107 37 128 133)(2 108 38 121 134)(3 109 39 122 135)(4 110 40 123 136)(5 111 33 124 129)(6 112 34 125 130)(7 105 35 126 131)(8 106 36 127 132)(9 138 75 58 98)(10 139 76 59 99)(11 140 77 60 100)(12 141 78 61 101)(13 142 79 62 102)(14 143 80 63 103)(15 144 73 64 104)(16 137 74 57 97)(17 47 56 119 30)(18 48 49 120 31)(19 41 50 113 32)(20 42 51 114 25)(21 43 52 115 26)(22 44 53 116 27)(23 45 54 117 28)(24 46 55 118 29)(65 150 153 87 94)(66 151 154 88 95)(67 152 155 81 96)(68 145 156 82 89)(69 146 157 83 90)(70 147 158 84 91)(71 148 159 85 92)(72 149 160 86 93)
(1 92)(2 89)(3 94)(4 91)(5 96)(6 93)(7 90)(8 95)(9 41)(10 46)(11 43)(12 48)(13 45)(14 42)(15 47)(16 44)(17 144)(18 141)(19 138)(20 143)(21 140)(22 137)(23 142)(24 139)(25 80)(26 77)(27 74)(28 79)(29 76)(30 73)(31 78)(32 75)(33 155)(34 160)(35 157)(36 154)(37 159)(38 156)(39 153)(40 158)(49 101)(50 98)(51 103)(52 100)(53 97)(54 102)(55 99)(56 104)(57 116)(58 113)(59 118)(60 115)(61 120)(62 117)(63 114)(64 119)(65 135)(66 132)(67 129)(68 134)(69 131)(70 136)(71 133)(72 130)(81 111)(82 108)(83 105)(84 110)(85 107)(86 112)(87 109)(88 106)(121 145)(122 150)(123 147)(124 152)(125 149)(126 146)(127 151)(128 148)
G:=sub<Sym(160)| (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,97)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,157)(18,158)(19,159)(20,160)(21,153)(22,154)(23,155)(24,156)(25,149)(26,150)(27,151)(28,152)(29,145)(30,146)(31,147)(32,148)(33,142)(34,143)(35,144)(36,137)(37,138)(38,139)(39,140)(40,141)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,89)(56,90)(57,132)(58,133)(59,134)(60,135)(61,136)(62,129)(63,130)(64,131)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125), (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), (1,67,5,71)(2,66,6,70)(3,65,7,69)(4,72,8,68)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,140,21,144)(18,139,22,143)(19,138,23,142)(20,137,24,141)(33,159,37,155)(34,158,38,154)(35,157,39,153)(36,156,40,160)(41,75,45,79)(42,74,46,78)(43,73,47,77)(44,80,48,76)(49,59,53,63)(50,58,54,62)(51,57,55,61)(52,64,56,60)(81,124,85,128)(82,123,86,127)(83,122,87,126)(84,121,88,125)(89,136,93,132)(90,135,94,131)(91,134,95,130)(92,133,96,129)(97,118,101,114)(98,117,102,113)(99,116,103,120)(100,115,104,119)(105,146,109,150)(106,145,110,149)(107,152,111,148)(108,151,112,147), (1,107,37,128,133)(2,108,38,121,134)(3,109,39,122,135)(4,110,40,123,136)(5,111,33,124,129)(6,112,34,125,130)(7,105,35,126,131)(8,106,36,127,132)(9,138,75,58,98)(10,139,76,59,99)(11,140,77,60,100)(12,141,78,61,101)(13,142,79,62,102)(14,143,80,63,103)(15,144,73,64,104)(16,137,74,57,97)(17,47,56,119,30)(18,48,49,120,31)(19,41,50,113,32)(20,42,51,114,25)(21,43,52,115,26)(22,44,53,116,27)(23,45,54,117,28)(24,46,55,118,29)(65,150,153,87,94)(66,151,154,88,95)(67,152,155,81,96)(68,145,156,82,89)(69,146,157,83,90)(70,147,158,84,91)(71,148,159,85,92)(72,149,160,86,93), (1,92)(2,89)(3,94)(4,91)(5,96)(6,93)(7,90)(8,95)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(17,144)(18,141)(19,138)(20,143)(21,140)(22,137)(23,142)(24,139)(25,80)(26,77)(27,74)(28,79)(29,76)(30,73)(31,78)(32,75)(33,155)(34,160)(35,157)(36,154)(37,159)(38,156)(39,153)(40,158)(49,101)(50,98)(51,103)(52,100)(53,97)(54,102)(55,99)(56,104)(57,116)(58,113)(59,118)(60,115)(61,120)(62,117)(63,114)(64,119)(65,135)(66,132)(67,129)(68,134)(69,131)(70,136)(71,133)(72,130)(81,111)(82,108)(83,105)(84,110)(85,107)(86,112)(87,109)(88,106)(121,145)(122,150)(123,147)(124,152)(125,149)(126,146)(127,151)(128,148)>;
G:=Group( (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,97)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,157)(18,158)(19,159)(20,160)(21,153)(22,154)(23,155)(24,156)(25,149)(26,150)(27,151)(28,152)(29,145)(30,146)(31,147)(32,148)(33,142)(34,143)(35,144)(36,137)(37,138)(38,139)(39,140)(40,141)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,89)(56,90)(57,132)(58,133)(59,134)(60,135)(61,136)(62,129)(63,130)(64,131)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125), (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), (1,67,5,71)(2,66,6,70)(3,65,7,69)(4,72,8,68)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,140,21,144)(18,139,22,143)(19,138,23,142)(20,137,24,141)(33,159,37,155)(34,158,38,154)(35,157,39,153)(36,156,40,160)(41,75,45,79)(42,74,46,78)(43,73,47,77)(44,80,48,76)(49,59,53,63)(50,58,54,62)(51,57,55,61)(52,64,56,60)(81,124,85,128)(82,123,86,127)(83,122,87,126)(84,121,88,125)(89,136,93,132)(90,135,94,131)(91,134,95,130)(92,133,96,129)(97,118,101,114)(98,117,102,113)(99,116,103,120)(100,115,104,119)(105,146,109,150)(106,145,110,149)(107,152,111,148)(108,151,112,147), (1,107,37,128,133)(2,108,38,121,134)(3,109,39,122,135)(4,110,40,123,136)(5,111,33,124,129)(6,112,34,125,130)(7,105,35,126,131)(8,106,36,127,132)(9,138,75,58,98)(10,139,76,59,99)(11,140,77,60,100)(12,141,78,61,101)(13,142,79,62,102)(14,143,80,63,103)(15,144,73,64,104)(16,137,74,57,97)(17,47,56,119,30)(18,48,49,120,31)(19,41,50,113,32)(20,42,51,114,25)(21,43,52,115,26)(22,44,53,116,27)(23,45,54,117,28)(24,46,55,118,29)(65,150,153,87,94)(66,151,154,88,95)(67,152,155,81,96)(68,145,156,82,89)(69,146,157,83,90)(70,147,158,84,91)(71,148,159,85,92)(72,149,160,86,93), (1,92)(2,89)(3,94)(4,91)(5,96)(6,93)(7,90)(8,95)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(17,144)(18,141)(19,138)(20,143)(21,140)(22,137)(23,142)(24,139)(25,80)(26,77)(27,74)(28,79)(29,76)(30,73)(31,78)(32,75)(33,155)(34,160)(35,157)(36,154)(37,159)(38,156)(39,153)(40,158)(49,101)(50,98)(51,103)(52,100)(53,97)(54,102)(55,99)(56,104)(57,116)(58,113)(59,118)(60,115)(61,120)(62,117)(63,114)(64,119)(65,135)(66,132)(67,129)(68,134)(69,131)(70,136)(71,133)(72,130)(81,111)(82,108)(83,105)(84,110)(85,107)(86,112)(87,109)(88,106)(121,145)(122,150)(123,147)(124,152)(125,149)(126,146)(127,151)(128,148) );
G=PermutationGroup([(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,97),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,105),(16,106),(17,157),(18,158),(19,159),(20,160),(21,153),(22,154),(23,155),(24,156),(25,149),(26,150),(27,151),(28,152),(29,145),(30,146),(31,147),(32,148),(33,142),(34,143),(35,144),(36,137),(37,138),(38,139),(39,140),(40,141),(41,85),(42,86),(43,87),(44,88),(45,81),(46,82),(47,83),(48,84),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,89),(56,90),(57,132),(58,133),(59,134),(60,135),(61,136),(62,129),(63,130),(64,131),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,113),(72,114),(73,126),(74,127),(75,128),(76,121),(77,122),(78,123),(79,124),(80,125)], [(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)], [(1,67,5,71),(2,66,6,70),(3,65,7,69),(4,72,8,68),(9,28,13,32),(10,27,14,31),(11,26,15,30),(12,25,16,29),(17,140,21,144),(18,139,22,143),(19,138,23,142),(20,137,24,141),(33,159,37,155),(34,158,38,154),(35,157,39,153),(36,156,40,160),(41,75,45,79),(42,74,46,78),(43,73,47,77),(44,80,48,76),(49,59,53,63),(50,58,54,62),(51,57,55,61),(52,64,56,60),(81,124,85,128),(82,123,86,127),(83,122,87,126),(84,121,88,125),(89,136,93,132),(90,135,94,131),(91,134,95,130),(92,133,96,129),(97,118,101,114),(98,117,102,113),(99,116,103,120),(100,115,104,119),(105,146,109,150),(106,145,110,149),(107,152,111,148),(108,151,112,147)], [(1,107,37,128,133),(2,108,38,121,134),(3,109,39,122,135),(4,110,40,123,136),(5,111,33,124,129),(6,112,34,125,130),(7,105,35,126,131),(8,106,36,127,132),(9,138,75,58,98),(10,139,76,59,99),(11,140,77,60,100),(12,141,78,61,101),(13,142,79,62,102),(14,143,80,63,103),(15,144,73,64,104),(16,137,74,57,97),(17,47,56,119,30),(18,48,49,120,31),(19,41,50,113,32),(20,42,51,114,25),(21,43,52,115,26),(22,44,53,116,27),(23,45,54,117,28),(24,46,55,118,29),(65,150,153,87,94),(66,151,154,88,95),(67,152,155,81,96),(68,145,156,82,89),(69,146,157,83,90),(70,147,158,84,91),(71,148,159,85,92),(72,149,160,86,93)], [(1,92),(2,89),(3,94),(4,91),(5,96),(6,93),(7,90),(8,95),(9,41),(10,46),(11,43),(12,48),(13,45),(14,42),(15,47),(16,44),(17,144),(18,141),(19,138),(20,143),(21,140),(22,137),(23,142),(24,139),(25,80),(26,77),(27,74),(28,79),(29,76),(30,73),(31,78),(32,75),(33,155),(34,160),(35,157),(36,154),(37,159),(38,156),(39,153),(40,158),(49,101),(50,98),(51,103),(52,100),(53,97),(54,102),(55,99),(56,104),(57,116),(58,113),(59,118),(60,115),(61,120),(62,117),(63,114),(64,119),(65,135),(66,132),(67,129),(68,134),(69,131),(70,136),(71,133),(72,130),(81,111),(82,108),(83,105),(84,110),(85,107),(86,112),(87,109),(88,106),(121,145),(122,150),(123,147),(124,152),(125,149),(126,146),(127,151),(128,148)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 17 | 2 | 24 |
| 0 | 0 | 24 | 2 | 17 | 39 |
| 0 | 0 | 40 | 29 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 22 | 0 |
| 0 | 0 | 1 | 13 | 0 | 22 |
| 0 | 0 | 17 | 40 | 35 | 1 |
| 0 | 0 | 1 | 24 | 40 | 28 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 36 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 35 | 10 | 19 |
| 0 | 0 | 8 | 2 | 31 | 31 |
| 0 | 0 | 36 | 11 | 8 | 13 |
| 0 | 0 | 5 | 5 | 39 | 33 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,39,24,40,12,0,0,17,2,29,1,0,0,2,17,0,0,0,0,24,39,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,1,17,1,0,0,40,13,40,24,0,0,22,0,35,40,0,0,0,22,1,28],[40,36,0,0,0,0,40,35,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[0,6,0,0,0,0,7,0,0,0,0,0,0,0,39,8,36,5,0,0,35,2,11,5,0,0,10,31,8,39,0,0,19,31,13,33] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C8.C22 | D4×D5 | D4×D5 | Q16⋊D5 |
| kernel | C2×Q16⋊D5 | C2×C8⋊D5 | C2×C40⋊C2 | Q16⋊D5 | C2×Q8⋊D5 | C2×C5⋊Q16 | C10×Q16 | C2×Q8×D5 | C2×Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times Q_{16}\rtimes D_5 % in TeX
G:=Group("C2xQ16:D5"); // GroupNames label
G:=SmallGroup(320,1436);
// by ID
G=gap.SmallGroup(320,1436);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,1123,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^5=e^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations
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