metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊9D20, (C2×C20)⋊19D4, C2.5(C4×D20), (C2×D20)⋊16C4, C10.32(C4×D4), C10.2C22≀C2, (C2×Dic5)⋊15D4, C22.59(D4×D5), D10⋊2(C22⋊C4), C10.3(C4⋊D4), C2.1(C4⋊D20), (C22×D20).1C2, C2.C42⋊7D5, (C22×C4).15D10, C22.24(C2×D20), C2.1(C22⋊D20), C2.2(D10⋊D4), C2.5(D20⋊8C4), (C22×D5).103D4, C5⋊2(C23.23D4), C22.34(C4○D20), (C23×D5).94C22, C23.255(C22×D5), C10.10C42⋊27C2, C2.3(D10.13D4), (C22×C10).290C23, (C22×C20).331C22, C22.17(Q8⋊2D5), C10.38(C22.D4), (C22×Dic5).14C22, (C2×C4)⋊2(C4×D5), (C2×C20)⋊17(C2×C4), (D5×C22×C4)⋊12C2, C2.7(D5×C22⋊C4), C22.89(C2×C4×D5), (C22×D5)⋊5(C2×C4), (C2×C10).199(C2×D4), C10.45(C2×C22⋊C4), (C2×D10⋊C4)⋊28C2, (C2×C10).183(C4○D4), (C5×C2.C42)⋊14C2, (C2×C10).150(C22×C4), SmallGroup(320,292)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊9D20
G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=dbd=ab=ba, ac=ca, ad=da, dcd=c-1 >
Subgroups: 1294 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×6], C4 [×8], C22 [×7], C22 [×26], C5, C2×C4 [×4], C2×C4 [×22], D4 [×8], C23, C23 [×18], D5 [×6], C10 [×7], C22⋊C4 [×6], C22×C4 [×3], C22×C4 [×8], C2×D4 [×8], C24 [×2], Dic5 [×3], C20 [×5], D10 [×4], D10 [×22], C2×C10 [×7], C2.C42, C2.C42, C2×C22⋊C4 [×3], C23×C4, C22×D4, C4×D5 [×8], D20 [×8], C2×Dic5 [×2], C2×Dic5 [×5], C2×C20 [×4], C2×C20 [×7], C22×D5 [×8], C22×D5 [×10], C22×C10, C23.23D4, D10⋊C4 [×6], C2×C4×D5 [×6], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C10.10C42, C5×C2.C42, C2×D10⋊C4 [×3], D5×C22×C4, C22×D20, (C2×C4)⋊9D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D5 [×2], D20 [×2], C22×D5, C23.23D4, C2×C4×D5, C2×D20, C4○D20, D4×D5 [×3], Q8⋊2D5, C4×D20, D5×C22⋊C4, C22⋊D20, D10⋊D4, D20⋊8C4, D10.13D4, C4⋊D20, (C2×C4)⋊9D20
►On 160 points
Generators in S160
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)(81 137)(82 138)(83 139)(84 140)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 158)(102 159)(103 160)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)
(1 103 121 27)(2 141 122 72)(3 105 123 29)(4 143 124 74)(5 107 125 31)(6 145 126 76)(7 109 127 33)(8 147 128 78)(9 111 129 35)(10 149 130 80)(11 113 131 37)(12 151 132 62)(13 115 133 39)(14 153 134 64)(15 117 135 21)(16 155 136 66)(17 119 137 23)(18 157 138 68)(19 101 139 25)(20 159 140 70)(22 44 118 100)(24 46 120 82)(26 48 102 84)(28 50 104 86)(30 52 106 88)(32 54 108 90)(34 56 110 92)(36 58 112 94)(38 60 114 96)(40 42 116 98)(41 152 97 63)(43 154 99 65)(45 156 81 67)(47 158 83 69)(49 160 85 71)(51 142 87 73)(53 144 89 75)(55 146 91 77)(57 148 93 79)(59 150 95 61)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 160)(118 159)(119 158)(120 157)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)
G:=sub<Sym(160)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(81,137)(82,138)(83,139)(84,140)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,158)(102,159)(103,160)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157), (1,103,121,27)(2,141,122,72)(3,105,123,29)(4,143,124,74)(5,107,125,31)(6,145,126,76)(7,109,127,33)(8,147,128,78)(9,111,129,35)(10,149,130,80)(11,113,131,37)(12,151,132,62)(13,115,133,39)(14,153,134,64)(15,117,135,21)(16,155,136,66)(17,119,137,23)(18,157,138,68)(19,101,139,25)(20,159,140,70)(22,44,118,100)(24,46,120,82)(26,48,102,84)(28,50,104,86)(30,52,106,88)(32,54,108,90)(34,56,110,92)(36,58,112,94)(38,60,114,96)(40,42,116,98)(41,152,97,63)(43,154,99,65)(45,156,81,67)(47,158,83,69)(49,160,85,71)(51,142,87,73)(53,144,89,75)(55,146,91,77)(57,148,93,79)(59,150,95,61), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,160)(118,159)(119,158)(120,157)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(81,137)(82,138)(83,139)(84,140)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,158)(102,159)(103,160)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157), (1,103,121,27)(2,141,122,72)(3,105,123,29)(4,143,124,74)(5,107,125,31)(6,145,126,76)(7,109,127,33)(8,147,128,78)(9,111,129,35)(10,149,130,80)(11,113,131,37)(12,151,132,62)(13,115,133,39)(14,153,134,64)(15,117,135,21)(16,155,136,66)(17,119,137,23)(18,157,138,68)(19,101,139,25)(20,159,140,70)(22,44,118,100)(24,46,120,82)(26,48,102,84)(28,50,104,86)(30,52,106,88)(32,54,108,90)(34,56,110,92)(36,58,112,94)(38,60,114,96)(40,42,116,98)(41,152,97,63)(43,154,99,65)(45,156,81,67)(47,158,83,69)(49,160,85,71)(51,142,87,73)(53,144,89,75)(55,146,91,77)(57,148,93,79)(59,150,95,61), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,160)(118,159)(119,158)(120,157)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139) );
G=PermutationGroup([(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64),(81,137),(82,138),(83,139),(84,140),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136),(101,158),(102,159),(103,160),(104,141),(105,142),(106,143),(107,144),(108,145),(109,146),(110,147),(111,148),(112,149),(113,150),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,157)], [(1,103,121,27),(2,141,122,72),(3,105,123,29),(4,143,124,74),(5,107,125,31),(6,145,126,76),(7,109,127,33),(8,147,128,78),(9,111,129,35),(10,149,130,80),(11,113,131,37),(12,151,132,62),(13,115,133,39),(14,153,134,64),(15,117,135,21),(16,155,136,66),(17,119,137,23),(18,157,138,68),(19,101,139,25),(20,159,140,70),(22,44,118,100),(24,46,120,82),(26,48,102,84),(28,50,104,86),(30,52,106,88),(32,54,108,90),(34,56,110,92),(36,58,112,94),(38,60,114,96),(40,42,116,98),(41,152,97,63),(43,154,99,65),(45,156,81,67),(47,158,83,69),(49,160,85,71),(51,142,87,73),(53,144,89,75),(55,146,91,77),(57,148,93,79),(59,150,95,61)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,156),(102,155),(103,154),(104,153),(105,152),(106,151),(107,150),(108,149),(109,148),(110,147),(111,146),(112,145),(113,144),(114,143),(115,142),(116,141),(117,160),(118,159),(119,158),(120,157),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,140),(137,139)])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D20 | C4○D20 | D4×D5 | Q8⋊2D5 |
| kernel | (C2×C4)⋊9D20 | C10.10C42 | C5×C2.C42 | C2×D10⋊C4 | D5×C22×C4 | C22×D20 | C2×D20 | C2×Dic5 | C2×C20 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 3 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 4 | 6 | 8 | 8 | 8 | 6 | 2 |
Matrix representation of (C2×C4)⋊9D20 ►in GL6(𝔽41)
| 1 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 30 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,30,0,0,0,0,11,14,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
(C2×C4)⋊9D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_9D_{20} % in TeX
G:=Group("(C2xC4):9D20"); // GroupNames label
G:=SmallGroup(320,292);
// by ID
G=gap.SmallGroup(320,292);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,d*c*d=c^-1>;
// generators/relations