metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.97D10, C10.982+ 1+4, C20⋊Q8⋊12C2, D10⋊D4⋊4C2, C20⋊7D4⋊30C2, C4⋊D20⋊13C2, C4⋊C4.311D10, C4.D20⋊7C2, C42⋊D5⋊4C2, D20⋊8C4⋊13C2, C4.97(C4○D20), C42⋊C2⋊16D5, (C4×C20).27C22, (C2×C10).76C24, C20.199(C4○D4), (C2×C20).697C23, C22⋊C4.100D10, Dic5.5D4⋊4C2, (C2×D20).26C22, (C22×C4).197D10, C2.10(D4⋊8D10), C23.87(C22×D5), Dic5.34(C4○D4), C4⋊Dic5.196C22, (C2×Dic5).29C23, (C22×D5).24C23, C22.105(C23×D5), C23.D5.98C22, D10⋊C4.83C22, (C22×C10).146C23, (C22×C20).233C22, C5⋊1(C22.49C24), (C4×Dic5).218C22, (C2×Dic10).26C22, C10.D4.106C22, (C4×C5⋊D4)⋊13C2, C2.15(D5×C4○D4), C4⋊C4⋊7D5⋊12C2, C2.35(C2×C4○D20), C10.32(C2×C4○D4), (C2×C4×D5).245C22, (C5×C42⋊C2)⋊18C2, (C5×C4⋊C4).312C22, (C2×C4).278(C22×D5), (C2×C5⋊D4).113C22, (C5×C22⋊C4).115C22, SmallGroup(320,1204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.97D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c9 >
Subgroups: 902 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.49C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C42⋊D5, C4.D20, D10⋊D4, Dic5.5D4, C20⋊Q8, C4⋊C4⋊7D5, D20⋊8C4, C4⋊D20, C4×C5⋊D4, C20⋊7D4, C5×C42⋊C2, C42.97D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.49C24, C4○D20, C23×D5, C2×C4○D20, D5×C4○D4, D4⋊8D10, C42.97D10
►On 160 points
Generators in S160
(1 78 113 121)(2 69 114 132)(3 80 115 123)(4 71 116 134)(5 62 117 125)(6 73 118 136)(7 64 119 127)(8 75 120 138)(9 66 101 129)(10 77 102 140)(11 68 103 131)(12 79 104 122)(13 70 105 133)(14 61 106 124)(15 72 107 135)(16 63 108 126)(17 74 109 137)(18 65 110 128)(19 76 111 139)(20 67 112 130)(21 147 92 43)(22 158 93 54)(23 149 94 45)(24 160 95 56)(25 151 96 47)(26 142 97 58)(27 153 98 49)(28 144 99 60)(29 155 100 51)(30 146 81 42)(31 157 82 53)(32 148 83 44)(33 159 84 55)(34 150 85 46)(35 141 86 57)(36 152 87 48)(37 143 88 59)(38 154 89 50)(39 145 90 41)(40 156 91 52)
(1 22 11 32)(2 23 12 33)(3 24 13 34)(4 25 14 35)(5 26 15 36)(6 27 16 37)(7 28 17 38)(8 29 18 39)(9 30 19 40)(10 31 20 21)(41 138 51 128)(42 139 52 129)(43 140 53 130)(44 121 54 131)(45 122 55 132)(46 123 56 133)(47 124 57 134)(48 125 58 135)(49 126 59 136)(50 127 60 137)(61 141 71 151)(62 142 72 152)(63 143 73 153)(64 144 74 154)(65 145 75 155)(66 146 76 156)(67 147 77 157)(68 148 78 158)(69 149 79 159)(70 150 80 160)(81 111 91 101)(82 112 92 102)(83 113 93 103)(84 114 94 104)(85 115 95 105)(86 116 96 106)(87 117 97 107)(88 118 98 108)(89 119 99 109)(90 120 100 110)
(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 48 11 58)(2 57 12 47)(3 46 13 56)(4 55 14 45)(5 44 15 54)(6 53 16 43)(7 42 17 52)(8 51 18 41)(9 60 19 50)(10 49 20 59)(21 73 31 63)(22 62 32 72)(23 71 33 61)(24 80 34 70)(25 69 35 79)(26 78 36 68)(27 67 37 77)(28 76 38 66)(29 65 39 75)(30 74 40 64)(81 137 91 127)(82 126 92 136)(83 135 93 125)(84 124 94 134)(85 133 95 123)(86 122 96 132)(87 131 97 121)(88 140 98 130)(89 129 99 139)(90 138 100 128)(101 144 111 154)(102 153 112 143)(103 142 113 152)(104 151 114 141)(105 160 115 150)(106 149 116 159)(107 158 117 148)(108 147 118 157)(109 156 119 146)(110 145 120 155)
G:=sub<Sym(160)| (1,78,113,121)(2,69,114,132)(3,80,115,123)(4,71,116,134)(5,62,117,125)(6,73,118,136)(7,64,119,127)(8,75,120,138)(9,66,101,129)(10,77,102,140)(11,68,103,131)(12,79,104,122)(13,70,105,133)(14,61,106,124)(15,72,107,135)(16,63,108,126)(17,74,109,137)(18,65,110,128)(19,76,111,139)(20,67,112,130)(21,147,92,43)(22,158,93,54)(23,149,94,45)(24,160,95,56)(25,151,96,47)(26,142,97,58)(27,153,98,49)(28,144,99,60)(29,155,100,51)(30,146,81,42)(31,157,82,53)(32,148,83,44)(33,159,84,55)(34,150,85,46)(35,141,86,57)(36,152,87,48)(37,143,88,59)(38,154,89,50)(39,145,90,41)(40,156,91,52), (1,22,11,32)(2,23,12,33)(3,24,13,34)(4,25,14,35)(5,26,15,36)(6,27,16,37)(7,28,17,38)(8,29,18,39)(9,30,19,40)(10,31,20,21)(41,138,51,128)(42,139,52,129)(43,140,53,130)(44,121,54,131)(45,122,55,132)(46,123,56,133)(47,124,57,134)(48,125,58,135)(49,126,59,136)(50,127,60,137)(61,141,71,151)(62,142,72,152)(63,143,73,153)(64,144,74,154)(65,145,75,155)(66,146,76,156)(67,147,77,157)(68,148,78,158)(69,149,79,159)(70,150,80,160)(81,111,91,101)(82,112,92,102)(83,113,93,103)(84,114,94,104)(85,115,95,105)(86,116,96,106)(87,117,97,107)(88,118,98,108)(89,119,99,109)(90,120,100,110), (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), (1,48,11,58)(2,57,12,47)(3,46,13,56)(4,55,14,45)(5,44,15,54)(6,53,16,43)(7,42,17,52)(8,51,18,41)(9,60,19,50)(10,49,20,59)(21,73,31,63)(22,62,32,72)(23,71,33,61)(24,80,34,70)(25,69,35,79)(26,78,36,68)(27,67,37,77)(28,76,38,66)(29,65,39,75)(30,74,40,64)(81,137,91,127)(82,126,92,136)(83,135,93,125)(84,124,94,134)(85,133,95,123)(86,122,96,132)(87,131,97,121)(88,140,98,130)(89,129,99,139)(90,138,100,128)(101,144,111,154)(102,153,112,143)(103,142,113,152)(104,151,114,141)(105,160,115,150)(106,149,116,159)(107,158,117,148)(108,147,118,157)(109,156,119,146)(110,145,120,155)>;
G:=Group( (1,78,113,121)(2,69,114,132)(3,80,115,123)(4,71,116,134)(5,62,117,125)(6,73,118,136)(7,64,119,127)(8,75,120,138)(9,66,101,129)(10,77,102,140)(11,68,103,131)(12,79,104,122)(13,70,105,133)(14,61,106,124)(15,72,107,135)(16,63,108,126)(17,74,109,137)(18,65,110,128)(19,76,111,139)(20,67,112,130)(21,147,92,43)(22,158,93,54)(23,149,94,45)(24,160,95,56)(25,151,96,47)(26,142,97,58)(27,153,98,49)(28,144,99,60)(29,155,100,51)(30,146,81,42)(31,157,82,53)(32,148,83,44)(33,159,84,55)(34,150,85,46)(35,141,86,57)(36,152,87,48)(37,143,88,59)(38,154,89,50)(39,145,90,41)(40,156,91,52), (1,22,11,32)(2,23,12,33)(3,24,13,34)(4,25,14,35)(5,26,15,36)(6,27,16,37)(7,28,17,38)(8,29,18,39)(9,30,19,40)(10,31,20,21)(41,138,51,128)(42,139,52,129)(43,140,53,130)(44,121,54,131)(45,122,55,132)(46,123,56,133)(47,124,57,134)(48,125,58,135)(49,126,59,136)(50,127,60,137)(61,141,71,151)(62,142,72,152)(63,143,73,153)(64,144,74,154)(65,145,75,155)(66,146,76,156)(67,147,77,157)(68,148,78,158)(69,149,79,159)(70,150,80,160)(81,111,91,101)(82,112,92,102)(83,113,93,103)(84,114,94,104)(85,115,95,105)(86,116,96,106)(87,117,97,107)(88,118,98,108)(89,119,99,109)(90,120,100,110), (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), (1,48,11,58)(2,57,12,47)(3,46,13,56)(4,55,14,45)(5,44,15,54)(6,53,16,43)(7,42,17,52)(8,51,18,41)(9,60,19,50)(10,49,20,59)(21,73,31,63)(22,62,32,72)(23,71,33,61)(24,80,34,70)(25,69,35,79)(26,78,36,68)(27,67,37,77)(28,76,38,66)(29,65,39,75)(30,74,40,64)(81,137,91,127)(82,126,92,136)(83,135,93,125)(84,124,94,134)(85,133,95,123)(86,122,96,132)(87,131,97,121)(88,140,98,130)(89,129,99,139)(90,138,100,128)(101,144,111,154)(102,153,112,143)(103,142,113,152)(104,151,114,141)(105,160,115,150)(106,149,116,159)(107,158,117,148)(108,147,118,157)(109,156,119,146)(110,145,120,155) );
G=PermutationGroup([[(1,78,113,121),(2,69,114,132),(3,80,115,123),(4,71,116,134),(5,62,117,125),(6,73,118,136),(7,64,119,127),(8,75,120,138),(9,66,101,129),(10,77,102,140),(11,68,103,131),(12,79,104,122),(13,70,105,133),(14,61,106,124),(15,72,107,135),(16,63,108,126),(17,74,109,137),(18,65,110,128),(19,76,111,139),(20,67,112,130),(21,147,92,43),(22,158,93,54),(23,149,94,45),(24,160,95,56),(25,151,96,47),(26,142,97,58),(27,153,98,49),(28,144,99,60),(29,155,100,51),(30,146,81,42),(31,157,82,53),(32,148,83,44),(33,159,84,55),(34,150,85,46),(35,141,86,57),(36,152,87,48),(37,143,88,59),(38,154,89,50),(39,145,90,41),(40,156,91,52)], [(1,22,11,32),(2,23,12,33),(3,24,13,34),(4,25,14,35),(5,26,15,36),(6,27,16,37),(7,28,17,38),(8,29,18,39),(9,30,19,40),(10,31,20,21),(41,138,51,128),(42,139,52,129),(43,140,53,130),(44,121,54,131),(45,122,55,132),(46,123,56,133),(47,124,57,134),(48,125,58,135),(49,126,59,136),(50,127,60,137),(61,141,71,151),(62,142,72,152),(63,143,73,153),(64,144,74,154),(65,145,75,155),(66,146,76,156),(67,147,77,157),(68,148,78,158),(69,149,79,159),(70,150,80,160),(81,111,91,101),(82,112,92,102),(83,113,93,103),(84,114,94,104),(85,115,95,105),(86,116,96,106),(87,117,97,107),(88,118,98,108),(89,119,99,109),(90,120,100,110)], [(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)], [(1,48,11,58),(2,57,12,47),(3,46,13,56),(4,55,14,45),(5,44,15,54),(6,53,16,43),(7,42,17,52),(8,51,18,41),(9,60,19,50),(10,49,20,59),(21,73,31,63),(22,62,32,72),(23,71,33,61),(24,80,34,70),(25,69,35,79),(26,78,36,68),(27,67,37,77),(28,76,38,66),(29,65,39,75),(30,74,40,64),(81,137,91,127),(82,126,92,136),(83,135,93,125),(84,124,94,134),(85,133,95,123),(86,122,96,132),(87,131,97,121),(88,140,98,130),(89,129,99,139),(90,138,100,128),(101,144,111,154),(102,153,112,143),(103,142,113,152),(104,151,114,141),(105,160,115,150),(106,149,116,159),(107,158,117,148),(108,147,118,157),(109,156,119,146),(110,145,120,155)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| 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 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C4○D20 | 2+ 1+4 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.97D10 | C42⋊D5 | C4.D20 | D10⋊D4 | Dic5.5D4 | C20⋊Q8 | C4⋊C4⋊7D5 | D20⋊8C4 | C4⋊D20 | C4×C5⋊D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | Dic5 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.97D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 7 |
| 0 | 0 | 0 | 0 | 18 | 23 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,39,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,18,0,0,0,0,7,23],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
C42.97D10 in GAP, Magma, Sage, TeX
C_4^2._{97}D_{10} % in TeX
G:=Group("C4^2.97D10"); // GroupNames label
G:=SmallGroup(320,1204);
// by ID
G=gap.SmallGroup(320,1204);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations