direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C4○D8, C40.79C23, C20.81C24, D8⋊6(C2×C10), (C2×D8)⋊13C10, (C10×D8)⋊27C2, (C22×C8)⋊8C10, Q16⋊6(C2×C10), C4.84(D4×C10), (C22×C40)⋊22C2, (C2×C40)⋊51C22, (C2×Q16)⋊13C10, (C10×Q16)⋊27C2, SD16⋊5(C2×C10), C20.328(C2×D4), (C2×C20).539D4, (C5×D8)⋊23C22, C4.4(C23×C10), C23.28(C5×D4), C22.4(D4×C10), (C10×SD16)⋊33C2, (C2×SD16)⋊16C10, C8.12(C22×C10), (C5×Q16)⋊20C22, D4.2(C22×C10), (C5×D4).35C23, (C5×Q8).36C23, Q8.2(C22×C10), (C2×C20).974C23, (C5×SD16)⋊22C22, (C22×C10).132D4, C10.202(C22×D4), (D4×C10).328C22, (Q8×C10).281C22, (C22×C20).604C22, C2.26(D4×C2×C10), (C2×C8)⋊13(C2×C10), C4○D4⋊3(C2×C10), (C10×C4○D4)⋊26C2, (C2×C4○D4)⋊10C10, (C2×C4).148(C5×D4), (C2×D4).74(C2×C10), (C2×C10).689(C2×D4), (C5×C4○D4)⋊23C22, (C2×Q8).69(C2×C10), (C22×C4).131(C2×C10), (C2×C4).144(C22×C10), SmallGroup(320,1574)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 266 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C10, C10 [×2], C10 [×6], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×10], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×10], C5×D4 [×4], C5×D4 [×10], C5×Q8 [×4], C5×Q8 [×2], C22×C10, C22×C10 [×2], C2×C4○D8, C2×C40 [×2], C2×C40 [×4], C5×D8 [×4], C5×SD16 [×8], C5×Q16 [×4], C22×C20, C22×C20 [×2], D4×C10 [×2], D4×C10 [×2], Q8×C10 [×2], C5×C4○D4 [×8], C5×C4○D4 [×4], C22×C40, C10×D8, C10×SD16 [×2], C10×Q16, C5×C4○D8 [×8], C10×C4○D4 [×2], C10×C4○D8
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C4○D8 [×2], C22×D4, C5×D4 [×4], C22×C10 [×15], C2×C4○D8, D4×C10 [×6], C23×C10, C5×C4○D8 [×2], D4×C2×C10, C10×C4○D8
Generators and relations
G = < a,b,c,d | a10=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
►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 99 58 79)(2 100 59 80)(3 91 60 71)(4 92 51 72)(5 93 52 73)(6 94 53 74)(7 95 54 75)(8 96 55 76)(9 97 56 77)(10 98 57 78)(11 121 29 143)(12 122 30 144)(13 123 21 145)(14 124 22 146)(15 125 23 147)(16 126 24 148)(17 127 25 149)(18 128 26 150)(19 129 27 141)(20 130 28 142)(31 113 158 133)(32 114 159 134)(33 115 160 135)(34 116 151 136)(35 117 152 137)(36 118 153 138)(37 119 154 139)(38 120 155 140)(39 111 156 131)(40 112 157 132)(41 86 63 108)(42 87 64 109)(43 88 65 110)(44 89 66 101)(45 90 67 102)(46 81 68 103)(47 82 69 104)(48 83 70 105)(49 84 61 106)(50 85 62 107)
(1 134 50 142 58 114 62 130)(2 135 41 143 59 115 63 121)(3 136 42 144 60 116 64 122)(4 137 43 145 51 117 65 123)(5 138 44 146 52 118 66 124)(6 139 45 147 53 119 67 125)(7 140 46 148 54 120 68 126)(8 131 47 149 55 111 69 127)(9 132 48 150 56 112 70 128)(10 133 49 141 57 113 61 129)(11 80 160 108 29 100 33 86)(12 71 151 109 30 91 34 87)(13 72 152 110 21 92 35 88)(14 73 153 101 22 93 36 89)(15 74 154 102 23 94 37 90)(16 75 155 103 24 95 38 81)(17 76 156 104 25 96 39 82)(18 77 157 105 26 97 40 83)(19 78 158 106 27 98 31 84)(20 79 159 107 28 99 32 85)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 38)(12 39)(13 40)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 157)(22 158)(23 159)(24 160)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(41 46)(42 47)(43 48)(44 49)(45 50)(61 66)(62 67)(63 68)(64 69)(65 70)(71 96)(72 97)(73 98)(74 99)(75 100)(76 91)(77 92)(78 93)(79 94)(80 95)(81 86)(82 87)(83 88)(84 89)(85 90)(101 106)(102 107)(103 108)(104 109)(105 110)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 121)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 141)(139 142)(140 143)
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), (1,99,58,79)(2,100,59,80)(3,91,60,71)(4,92,51,72)(5,93,52,73)(6,94,53,74)(7,95,54,75)(8,96,55,76)(9,97,56,77)(10,98,57,78)(11,121,29,143)(12,122,30,144)(13,123,21,145)(14,124,22,146)(15,125,23,147)(16,126,24,148)(17,127,25,149)(18,128,26,150)(19,129,27,141)(20,130,28,142)(31,113,158,133)(32,114,159,134)(33,115,160,135)(34,116,151,136)(35,117,152,137)(36,118,153,138)(37,119,154,139)(38,120,155,140)(39,111,156,131)(40,112,157,132)(41,86,63,108)(42,87,64,109)(43,88,65,110)(44,89,66,101)(45,90,67,102)(46,81,68,103)(47,82,69,104)(48,83,70,105)(49,84,61,106)(50,85,62,107), (1,134,50,142,58,114,62,130)(2,135,41,143,59,115,63,121)(3,136,42,144,60,116,64,122)(4,137,43,145,51,117,65,123)(5,138,44,146,52,118,66,124)(6,139,45,147,53,119,67,125)(7,140,46,148,54,120,68,126)(8,131,47,149,55,111,69,127)(9,132,48,150,56,112,70,128)(10,133,49,141,57,113,61,129)(11,80,160,108,29,100,33,86)(12,71,151,109,30,91,34,87)(13,72,152,110,21,92,35,88)(14,73,153,101,22,93,36,89)(15,74,154,102,23,94,37,90)(16,75,155,103,24,95,38,81)(17,76,156,104,25,96,39,82)(18,77,157,105,26,97,40,83)(19,78,158,106,27,98,31,84)(20,79,159,107,28,99,32,85), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,38)(12,39)(13,40)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,157)(22,158)(23,159)(24,160)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(41,46)(42,47)(43,48)(44,49)(45,50)(61,66)(62,67)(63,68)(64,69)(65,70)(71,96)(72,97)(73,98)(74,99)(75,100)(76,91)(77,92)(78,93)(79,94)(80,95)(81,86)(82,87)(83,88)(84,89)(85,90)(101,106)(102,107)(103,108)(104,109)(105,110)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,121)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,141)(139,142)(140,143)>;
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), (1,99,58,79)(2,100,59,80)(3,91,60,71)(4,92,51,72)(5,93,52,73)(6,94,53,74)(7,95,54,75)(8,96,55,76)(9,97,56,77)(10,98,57,78)(11,121,29,143)(12,122,30,144)(13,123,21,145)(14,124,22,146)(15,125,23,147)(16,126,24,148)(17,127,25,149)(18,128,26,150)(19,129,27,141)(20,130,28,142)(31,113,158,133)(32,114,159,134)(33,115,160,135)(34,116,151,136)(35,117,152,137)(36,118,153,138)(37,119,154,139)(38,120,155,140)(39,111,156,131)(40,112,157,132)(41,86,63,108)(42,87,64,109)(43,88,65,110)(44,89,66,101)(45,90,67,102)(46,81,68,103)(47,82,69,104)(48,83,70,105)(49,84,61,106)(50,85,62,107), (1,134,50,142,58,114,62,130)(2,135,41,143,59,115,63,121)(3,136,42,144,60,116,64,122)(4,137,43,145,51,117,65,123)(5,138,44,146,52,118,66,124)(6,139,45,147,53,119,67,125)(7,140,46,148,54,120,68,126)(8,131,47,149,55,111,69,127)(9,132,48,150,56,112,70,128)(10,133,49,141,57,113,61,129)(11,80,160,108,29,100,33,86)(12,71,151,109,30,91,34,87)(13,72,152,110,21,92,35,88)(14,73,153,101,22,93,36,89)(15,74,154,102,23,94,37,90)(16,75,155,103,24,95,38,81)(17,76,156,104,25,96,39,82)(18,77,157,105,26,97,40,83)(19,78,158,106,27,98,31,84)(20,79,159,107,28,99,32,85), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,38)(12,39)(13,40)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,157)(22,158)(23,159)(24,160)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(41,46)(42,47)(43,48)(44,49)(45,50)(61,66)(62,67)(63,68)(64,69)(65,70)(71,96)(72,97)(73,98)(74,99)(75,100)(76,91)(77,92)(78,93)(79,94)(80,95)(81,86)(82,87)(83,88)(84,89)(85,90)(101,106)(102,107)(103,108)(104,109)(105,110)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,121)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,141)(139,142)(140,143) );
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)], [(1,99,58,79),(2,100,59,80),(3,91,60,71),(4,92,51,72),(5,93,52,73),(6,94,53,74),(7,95,54,75),(8,96,55,76),(9,97,56,77),(10,98,57,78),(11,121,29,143),(12,122,30,144),(13,123,21,145),(14,124,22,146),(15,125,23,147),(16,126,24,148),(17,127,25,149),(18,128,26,150),(19,129,27,141),(20,130,28,142),(31,113,158,133),(32,114,159,134),(33,115,160,135),(34,116,151,136),(35,117,152,137),(36,118,153,138),(37,119,154,139),(38,120,155,140),(39,111,156,131),(40,112,157,132),(41,86,63,108),(42,87,64,109),(43,88,65,110),(44,89,66,101),(45,90,67,102),(46,81,68,103),(47,82,69,104),(48,83,70,105),(49,84,61,106),(50,85,62,107)], [(1,134,50,142,58,114,62,130),(2,135,41,143,59,115,63,121),(3,136,42,144,60,116,64,122),(4,137,43,145,51,117,65,123),(5,138,44,146,52,118,66,124),(6,139,45,147,53,119,67,125),(7,140,46,148,54,120,68,126),(8,131,47,149,55,111,69,127),(9,132,48,150,56,112,70,128),(10,133,49,141,57,113,61,129),(11,80,160,108,29,100,33,86),(12,71,151,109,30,91,34,87),(13,72,152,110,21,92,35,88),(14,73,153,101,22,93,36,89),(15,74,154,102,23,94,37,90),(16,75,155,103,24,95,38,81),(17,76,156,104,25,96,39,82),(18,77,157,105,26,97,40,83),(19,78,158,106,27,98,31,84),(20,79,159,107,28,99,32,85)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,51),(10,52),(11,38),(12,39),(13,40),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,157),(22,158),(23,159),(24,160),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(41,46),(42,47),(43,48),(44,49),(45,50),(61,66),(62,67),(63,68),(64,69),(65,70),(71,96),(72,97),(73,98),(74,99),(75,100),(76,91),(77,92),(78,93),(79,94),(80,95),(81,86),(82,87),(83,88),(84,89),(85,90),(101,106),(102,107),(103,108),(104,109),(105,110),(111,122),(112,123),(113,124),(114,125),(115,126),(116,127),(117,128),(118,129),(119,130),(120,121),(131,144),(132,145),(133,146),(134,147),(135,148),(136,149),(137,150),(138,141),(139,142),(140,143)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 29 | 29 |
| 40 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [23,0,0,0,0,23,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[1,40,0,0,2,40,0,0,0,0,29,29,0,0,12,29],[40,1,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D8 | C5×D4 | C5×D4 | C5×C4○D8 |
| kernel | C10×C4○D8 | C22×C40 | C10×D8 | C10×SD16 | C10×Q16 | C5×C4○D8 | C10×C4○D4 | C2×C4○D8 | C22×C8 | C2×D8 | C2×SD16 | C2×Q16 | C4○D8 | C2×C4○D4 | C2×C20 | C22×C10 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 4 | 4 | 4 | 8 | 4 | 32 | 8 | 3 | 1 | 8 | 12 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4\circ D_8 % in TeX
G:=Group("C10xC4oD8"); // GroupNames label
G:=SmallGroup(320,1574);
// by ID
G=gap.SmallGroup(320,1574);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,856,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀