metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.143D10, C10.1272+ (1+4), (C4×D20)⋊46C2, (Q8×Dic5)⋊20C2, (D4×Dic5)⋊31C2, C4.4D4⋊14D5, (C4×Dic10)⋊46C2, (C2×D4).176D10, C20⋊D4.11C2, (C2×Q8).139D10, C22⋊C4.36D10, Dic5⋊4D4⋊34C2, C20.126(C4○D4), C4.16(D4⋊2D5), C20.23D4⋊23C2, (C2×C20).505C23, (C2×C10).225C24, (C4×C20).188C22, D10.12D4⋊46C2, C2.51(D4⋊8D10), C23.47(C22×D5), Dic5.65(C4○D4), Dic5.5D4⋊41C2, (D4×C10).158C22, (C2×D20).274C22, C22.D20⋊26C2, C4⋊Dic5.235C22, (C22×C10).55C23, (Q8×C10).129C22, (C22×D5).97C23, C22.246(C23×D5), C23.D5.58C22, D10⋊C4.37C22, C5⋊4(C22.53C24), (C2×Dic5).266C23, (C4×Dic5).143C22, (C2×Dic10).258C22, C10.D4.142C22, (C22×Dic5).145C22, C2.81(D5×C4○D4), C10.192(C2×C4○D4), C2.57(C2×D4⋊2D5), (C5×C4.4D4)⋊17C2, (C2×C4×D5).267C22, (C2×C4).198(C22×D5), (C2×C5⋊D4).63C22, (C5×C22⋊C4).67C22, SmallGroup(320,1353)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 854 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×6], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4, C4.4D4 [×3], C4⋊1D4, Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], C22.53C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic5⋊4D4 [×2], D10.12D4 [×2], Dic5.5D4 [×2], C22.D20 [×2], D4×Dic5, C20⋊D4, Q8×Dic5, C20.23D4, C5×C4.4D4, C42.143D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.53C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D5×C4○D4, D4⋊8D10, C42.143D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
►On 160 points
Generators in S160
(1 110 63 114)(2 144 64 90)(3 102 65 116)(4 146 66 82)(5 104 67 118)(6 148 68 84)(7 106 69 120)(8 150 70 86)(9 108 61 112)(10 142 62 88)(11 43 97 129)(12 151 98 135)(13 45 99 121)(14 153 100 137)(15 47 91 123)(16 155 92 139)(17 49 93 125)(18 157 94 131)(19 41 95 127)(20 159 96 133)(21 160 80 134)(22 44 71 130)(23 152 72 136)(24 46 73 122)(25 154 74 138)(26 48 75 124)(27 156 76 140)(28 50 77 126)(29 158 78 132)(30 42 79 128)(31 115 58 101)(32 81 59 145)(33 117 60 103)(34 83 51 147)(35 119 52 105)(36 85 53 149)(37 111 54 107)(38 87 55 141)(39 113 56 109)(40 89 57 143)
(1 14 57 24)(2 91 58 74)(3 16 59 26)(4 93 60 76)(5 18 51 28)(6 95 52 78)(7 20 53 30)(8 97 54 80)(9 12 55 22)(10 99 56 72)(11 37 21 70)(13 39 23 62)(15 31 25 64)(17 33 27 66)(19 35 29 68)(32 75 65 92)(34 77 67 94)(36 79 69 96)(38 71 61 98)(40 73 63 100)(41 119 158 84)(42 106 159 149)(43 111 160 86)(44 108 151 141)(45 113 152 88)(46 110 153 143)(47 115 154 90)(48 102 155 145)(49 117 156 82)(50 104 157 147)(81 124 116 139)(83 126 118 131)(85 128 120 133)(87 130 112 135)(89 122 114 137)(101 138 144 123)(103 140 146 125)(105 132 148 127)(107 134 150 129)(109 136 142 121)
(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 10 63 62)(2 61 64 9)(3 8 65 70)(4 69 66 7)(5 6 67 68)(11 26 97 75)(12 74 98 25)(13 24 99 73)(14 72 100 23)(15 22 91 71)(16 80 92 21)(17 30 93 79)(18 78 94 29)(19 28 95 77)(20 76 96 27)(31 55 58 38)(32 37 59 54)(33 53 60 36)(34 35 51 52)(39 57 56 40)(41 157 127 131)(42 140 128 156)(43 155 129 139)(44 138 130 154)(45 153 121 137)(46 136 122 152)(47 151 123 135)(48 134 124 160)(49 159 125 133)(50 132 126 158)(81 86 145 150)(82 149 146 85)(83 84 147 148)(87 90 141 144)(88 143 142 89)(101 112 115 108)(102 107 116 111)(103 120 117 106)(104 105 118 119)(109 114 113 110)
G:=sub<Sym(160)| (1,110,63,114)(2,144,64,90)(3,102,65,116)(4,146,66,82)(5,104,67,118)(6,148,68,84)(7,106,69,120)(8,150,70,86)(9,108,61,112)(10,142,62,88)(11,43,97,129)(12,151,98,135)(13,45,99,121)(14,153,100,137)(15,47,91,123)(16,155,92,139)(17,49,93,125)(18,157,94,131)(19,41,95,127)(20,159,96,133)(21,160,80,134)(22,44,71,130)(23,152,72,136)(24,46,73,122)(25,154,74,138)(26,48,75,124)(27,156,76,140)(28,50,77,126)(29,158,78,132)(30,42,79,128)(31,115,58,101)(32,81,59,145)(33,117,60,103)(34,83,51,147)(35,119,52,105)(36,85,53,149)(37,111,54,107)(38,87,55,141)(39,113,56,109)(40,89,57,143), (1,14,57,24)(2,91,58,74)(3,16,59,26)(4,93,60,76)(5,18,51,28)(6,95,52,78)(7,20,53,30)(8,97,54,80)(9,12,55,22)(10,99,56,72)(11,37,21,70)(13,39,23,62)(15,31,25,64)(17,33,27,66)(19,35,29,68)(32,75,65,92)(34,77,67,94)(36,79,69,96)(38,71,61,98)(40,73,63,100)(41,119,158,84)(42,106,159,149)(43,111,160,86)(44,108,151,141)(45,113,152,88)(46,110,153,143)(47,115,154,90)(48,102,155,145)(49,117,156,82)(50,104,157,147)(81,124,116,139)(83,126,118,131)(85,128,120,133)(87,130,112,135)(89,122,114,137)(101,138,144,123)(103,140,146,125)(105,132,148,127)(107,134,150,129)(109,136,142,121), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,26,97,75)(12,74,98,25)(13,24,99,73)(14,72,100,23)(15,22,91,71)(16,80,92,21)(17,30,93,79)(18,78,94,29)(19,28,95,77)(20,76,96,27)(31,55,58,38)(32,37,59,54)(33,53,60,36)(34,35,51,52)(39,57,56,40)(41,157,127,131)(42,140,128,156)(43,155,129,139)(44,138,130,154)(45,153,121,137)(46,136,122,152)(47,151,123,135)(48,134,124,160)(49,159,125,133)(50,132,126,158)(81,86,145,150)(82,149,146,85)(83,84,147,148)(87,90,141,144)(88,143,142,89)(101,112,115,108)(102,107,116,111)(103,120,117,106)(104,105,118,119)(109,114,113,110)>;
G:=Group( (1,110,63,114)(2,144,64,90)(3,102,65,116)(4,146,66,82)(5,104,67,118)(6,148,68,84)(7,106,69,120)(8,150,70,86)(9,108,61,112)(10,142,62,88)(11,43,97,129)(12,151,98,135)(13,45,99,121)(14,153,100,137)(15,47,91,123)(16,155,92,139)(17,49,93,125)(18,157,94,131)(19,41,95,127)(20,159,96,133)(21,160,80,134)(22,44,71,130)(23,152,72,136)(24,46,73,122)(25,154,74,138)(26,48,75,124)(27,156,76,140)(28,50,77,126)(29,158,78,132)(30,42,79,128)(31,115,58,101)(32,81,59,145)(33,117,60,103)(34,83,51,147)(35,119,52,105)(36,85,53,149)(37,111,54,107)(38,87,55,141)(39,113,56,109)(40,89,57,143), (1,14,57,24)(2,91,58,74)(3,16,59,26)(4,93,60,76)(5,18,51,28)(6,95,52,78)(7,20,53,30)(8,97,54,80)(9,12,55,22)(10,99,56,72)(11,37,21,70)(13,39,23,62)(15,31,25,64)(17,33,27,66)(19,35,29,68)(32,75,65,92)(34,77,67,94)(36,79,69,96)(38,71,61,98)(40,73,63,100)(41,119,158,84)(42,106,159,149)(43,111,160,86)(44,108,151,141)(45,113,152,88)(46,110,153,143)(47,115,154,90)(48,102,155,145)(49,117,156,82)(50,104,157,147)(81,124,116,139)(83,126,118,131)(85,128,120,133)(87,130,112,135)(89,122,114,137)(101,138,144,123)(103,140,146,125)(105,132,148,127)(107,134,150,129)(109,136,142,121), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,26,97,75)(12,74,98,25)(13,24,99,73)(14,72,100,23)(15,22,91,71)(16,80,92,21)(17,30,93,79)(18,78,94,29)(19,28,95,77)(20,76,96,27)(31,55,58,38)(32,37,59,54)(33,53,60,36)(34,35,51,52)(39,57,56,40)(41,157,127,131)(42,140,128,156)(43,155,129,139)(44,138,130,154)(45,153,121,137)(46,136,122,152)(47,151,123,135)(48,134,124,160)(49,159,125,133)(50,132,126,158)(81,86,145,150)(82,149,146,85)(83,84,147,148)(87,90,141,144)(88,143,142,89)(101,112,115,108)(102,107,116,111)(103,120,117,106)(104,105,118,119)(109,114,113,110) );
G=PermutationGroup([(1,110,63,114),(2,144,64,90),(3,102,65,116),(4,146,66,82),(5,104,67,118),(6,148,68,84),(7,106,69,120),(8,150,70,86),(9,108,61,112),(10,142,62,88),(11,43,97,129),(12,151,98,135),(13,45,99,121),(14,153,100,137),(15,47,91,123),(16,155,92,139),(17,49,93,125),(18,157,94,131),(19,41,95,127),(20,159,96,133),(21,160,80,134),(22,44,71,130),(23,152,72,136),(24,46,73,122),(25,154,74,138),(26,48,75,124),(27,156,76,140),(28,50,77,126),(29,158,78,132),(30,42,79,128),(31,115,58,101),(32,81,59,145),(33,117,60,103),(34,83,51,147),(35,119,52,105),(36,85,53,149),(37,111,54,107),(38,87,55,141),(39,113,56,109),(40,89,57,143)], [(1,14,57,24),(2,91,58,74),(3,16,59,26),(4,93,60,76),(5,18,51,28),(6,95,52,78),(7,20,53,30),(8,97,54,80),(9,12,55,22),(10,99,56,72),(11,37,21,70),(13,39,23,62),(15,31,25,64),(17,33,27,66),(19,35,29,68),(32,75,65,92),(34,77,67,94),(36,79,69,96),(38,71,61,98),(40,73,63,100),(41,119,158,84),(42,106,159,149),(43,111,160,86),(44,108,151,141),(45,113,152,88),(46,110,153,143),(47,115,154,90),(48,102,155,145),(49,117,156,82),(50,104,157,147),(81,124,116,139),(83,126,118,131),(85,128,120,133),(87,130,112,135),(89,122,114,137),(101,138,144,123),(103,140,146,125),(105,132,148,127),(107,134,150,129),(109,136,142,121)], [(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)], [(1,10,63,62),(2,61,64,9),(3,8,65,70),(4,69,66,7),(5,6,67,68),(11,26,97,75),(12,74,98,25),(13,24,99,73),(14,72,100,23),(15,22,91,71),(16,80,92,21),(17,30,93,79),(18,78,94,29),(19,28,95,77),(20,76,96,27),(31,55,58,38),(32,37,59,54),(33,53,60,36),(34,35,51,52),(39,57,56,40),(41,157,127,131),(42,140,128,156),(43,155,129,139),(44,138,130,154),(45,153,121,137),(46,136,122,152),(47,151,123,135),(48,134,124,160),(49,159,125,133),(50,132,126,158),(81,86,145,150),(82,149,146,85),(83,84,147,148),(87,90,141,144),(88,143,142,89),(101,112,115,108),(102,107,116,111),(103,120,117,106),(104,105,118,119),(109,114,113,110)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 9 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 |
| 32 | 16 | 0 | 0 | 0 | 0 |
| 36 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 |
| 40 | 20 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 8 |
| 0 | 0 | 0 | 0 | 15 | 39 |
| 40 | 20 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 33 |
| 0 | 0 | 0 | 0 | 16 | 2 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[32,36,0,0,0,0,16,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[40,0,0,0,0,0,20,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,2,15,0,0,0,0,8,39],[40,4,0,0,0,0,20,1,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,39,16,0,0,0,0,33,2] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.143D10 | C4×Dic10 | C4×D20 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C22.D20 | D4×Dic5 | C20⋊D4 | Q8×Dic5 | C20.23D4 | C5×C4.4D4 | C4.4D4 | Dic5 | C20 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{143}D_{10} % in TeX
G:=Group("C4^2.143D10"); // GroupNames label
G:=SmallGroup(320,1353);
// by ID
G=gap.SmallGroup(320,1353);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀