metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.189D10, C4⋊C4.212D10, (D5×C42)⋊20C2, D20⋊8C4⋊41C2, D10⋊Q8⋊42C2, D10⋊D4.3C2, C42⋊2C2⋊10D5, C42⋊2D5⋊14C2, (C2×C20).95C23, C22⋊C4.78D10, Dic5⋊4D4⋊35C2, Dic5⋊3Q8⋊41C2, D10.19(C4○D4), (C2×C10).250C24, (C4×C20).234C22, D10.13D4⋊40C2, D10.12D4⋊50C2, C23.56(C22×D5), Dic5.21(C4○D4), Dic5.Q8⋊37C2, Dic5.5D4⋊46C2, (C2×D20).174C22, C4⋊Dic5.246C22, (C22×C10).64C23, C22.271(C23×D5), C23.D5.66C22, D10⋊C4.45C22, C23.11D10⋊21C2, (C4×Dic5).158C22, (C2×Dic5).380C23, C10.D4.72C22, (C22×D5).234C23, C5⋊11(C23.36C23), (C2×Dic10).189C22, (C22×Dic5).150C22, C2.97(D5×C4○D4), C4⋊C4⋊7D5⋊40C2, (C5×C42⋊2C2)⋊5C2, C10.208(C2×C4○D4), (C2×C4×D5).383C22, (C2×C4).87(C22×D5), (C5×C4⋊C4).202C22, (C2×C5⋊D4).70C22, (C5×C22⋊C4).75C22, SmallGroup(320,1378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.189D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
Subgroups: 798 in 234 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C23.36C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, D5×C42, C42⋊2D5, C23.11D10, Dic5⋊4D4, D10.12D4, D10⋊D4, Dic5.5D4, Dic5⋊3Q8, Dic5.Q8, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4, D10⋊Q8, C5×C42⋊2C2, C42.189D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, C23×D5, D5×C4○D4, C42.189D10
►On 160 points
Generators in S160
(1 108 33 140)(2 114 34 81)(3 110 35 132)(4 116 36 83)(5 102 37 134)(6 118 38 85)(7 104 39 136)(8 120 40 87)(9 106 31 138)(10 112 32 89)(11 131 46 109)(12 82 47 115)(13 133 48 101)(14 84 49 117)(15 135 50 103)(16 86 41 119)(17 137 42 105)(18 88 43 111)(19 139 44 107)(20 90 45 113)(21 100 149 55)(22 64 150 79)(23 92 141 57)(24 66 142 71)(25 94 143 59)(26 68 144 73)(27 96 145 51)(28 70 146 75)(29 98 147 53)(30 62 148 77)(52 123 97 155)(54 125 99 157)(56 127 91 159)(58 129 93 151)(60 121 95 153)(61 156 76 124)(63 158 78 126)(65 160 80 128)(67 152 72 130)(69 154 74 122)
(1 95 20 73)(2 51 11 69)(3 97 12 75)(4 53 13 61)(5 99 14 77)(6 55 15 63)(7 91 16 79)(8 57 17 65)(9 93 18 71)(10 59 19 67)(21 135 158 118)(22 104 159 86)(23 137 160 120)(24 106 151 88)(25 139 152 112)(26 108 153 90)(27 131 154 114)(28 110 155 82)(29 133 156 116)(30 102 157 84)(31 58 43 66)(32 94 44 72)(33 60 45 68)(34 96 46 74)(35 52 47 70)(36 98 48 76)(37 54 49 62)(38 100 50 78)(39 56 41 64)(40 92 42 80)(81 145 109 122)(83 147 101 124)(85 149 103 126)(87 141 105 128)(89 143 107 130)(111 142 138 129)(113 144 140 121)(115 146 132 123)(117 148 134 125)(119 150 136 127)
(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)
(1 125 33 157)(2 156 34 124)(3 123 35 155)(4 154 36 122)(5 121 37 153)(6 152 38 130)(7 129 39 151)(8 160 40 128)(9 127 31 159)(10 158 32 126)(11 29 46 147)(12 146 47 28)(13 27 48 145)(14 144 49 26)(15 25 50 143)(16 142 41 24)(17 23 42 141)(18 150 43 22)(19 21 44 149)(20 148 45 30)(51 83 96 116)(52 115 97 82)(53 81 98 114)(54 113 99 90)(55 89 100 112)(56 111 91 88)(57 87 92 120)(58 119 93 86)(59 85 94 118)(60 117 95 84)(61 109 76 131)(62 140 77 108)(63 107 78 139)(64 138 79 106)(65 105 80 137)(66 136 71 104)(67 103 72 135)(68 134 73 102)(69 101 74 133)(70 132 75 110)
G:=sub<Sym(160)| (1,108,33,140)(2,114,34,81)(3,110,35,132)(4,116,36,83)(5,102,37,134)(6,118,38,85)(7,104,39,136)(8,120,40,87)(9,106,31,138)(10,112,32,89)(11,131,46,109)(12,82,47,115)(13,133,48,101)(14,84,49,117)(15,135,50,103)(16,86,41,119)(17,137,42,105)(18,88,43,111)(19,139,44,107)(20,90,45,113)(21,100,149,55)(22,64,150,79)(23,92,141,57)(24,66,142,71)(25,94,143,59)(26,68,144,73)(27,96,145,51)(28,70,146,75)(29,98,147,53)(30,62,148,77)(52,123,97,155)(54,125,99,157)(56,127,91,159)(58,129,93,151)(60,121,95,153)(61,156,76,124)(63,158,78,126)(65,160,80,128)(67,152,72,130)(69,154,74,122), (1,95,20,73)(2,51,11,69)(3,97,12,75)(4,53,13,61)(5,99,14,77)(6,55,15,63)(7,91,16,79)(8,57,17,65)(9,93,18,71)(10,59,19,67)(21,135,158,118)(22,104,159,86)(23,137,160,120)(24,106,151,88)(25,139,152,112)(26,108,153,90)(27,131,154,114)(28,110,155,82)(29,133,156,116)(30,102,157,84)(31,58,43,66)(32,94,44,72)(33,60,45,68)(34,96,46,74)(35,52,47,70)(36,98,48,76)(37,54,49,62)(38,100,50,78)(39,56,41,64)(40,92,42,80)(81,145,109,122)(83,147,101,124)(85,149,103,126)(87,141,105,128)(89,143,107,130)(111,142,138,129)(113,144,140,121)(115,146,132,123)(117,148,134,125)(119,150,136,127), (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), (1,125,33,157)(2,156,34,124)(3,123,35,155)(4,154,36,122)(5,121,37,153)(6,152,38,130)(7,129,39,151)(8,160,40,128)(9,127,31,159)(10,158,32,126)(11,29,46,147)(12,146,47,28)(13,27,48,145)(14,144,49,26)(15,25,50,143)(16,142,41,24)(17,23,42,141)(18,150,43,22)(19,21,44,149)(20,148,45,30)(51,83,96,116)(52,115,97,82)(53,81,98,114)(54,113,99,90)(55,89,100,112)(56,111,91,88)(57,87,92,120)(58,119,93,86)(59,85,94,118)(60,117,95,84)(61,109,76,131)(62,140,77,108)(63,107,78,139)(64,138,79,106)(65,105,80,137)(66,136,71,104)(67,103,72,135)(68,134,73,102)(69,101,74,133)(70,132,75,110)>;
G:=Group( (1,108,33,140)(2,114,34,81)(3,110,35,132)(4,116,36,83)(5,102,37,134)(6,118,38,85)(7,104,39,136)(8,120,40,87)(9,106,31,138)(10,112,32,89)(11,131,46,109)(12,82,47,115)(13,133,48,101)(14,84,49,117)(15,135,50,103)(16,86,41,119)(17,137,42,105)(18,88,43,111)(19,139,44,107)(20,90,45,113)(21,100,149,55)(22,64,150,79)(23,92,141,57)(24,66,142,71)(25,94,143,59)(26,68,144,73)(27,96,145,51)(28,70,146,75)(29,98,147,53)(30,62,148,77)(52,123,97,155)(54,125,99,157)(56,127,91,159)(58,129,93,151)(60,121,95,153)(61,156,76,124)(63,158,78,126)(65,160,80,128)(67,152,72,130)(69,154,74,122), (1,95,20,73)(2,51,11,69)(3,97,12,75)(4,53,13,61)(5,99,14,77)(6,55,15,63)(7,91,16,79)(8,57,17,65)(9,93,18,71)(10,59,19,67)(21,135,158,118)(22,104,159,86)(23,137,160,120)(24,106,151,88)(25,139,152,112)(26,108,153,90)(27,131,154,114)(28,110,155,82)(29,133,156,116)(30,102,157,84)(31,58,43,66)(32,94,44,72)(33,60,45,68)(34,96,46,74)(35,52,47,70)(36,98,48,76)(37,54,49,62)(38,100,50,78)(39,56,41,64)(40,92,42,80)(81,145,109,122)(83,147,101,124)(85,149,103,126)(87,141,105,128)(89,143,107,130)(111,142,138,129)(113,144,140,121)(115,146,132,123)(117,148,134,125)(119,150,136,127), (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), (1,125,33,157)(2,156,34,124)(3,123,35,155)(4,154,36,122)(5,121,37,153)(6,152,38,130)(7,129,39,151)(8,160,40,128)(9,127,31,159)(10,158,32,126)(11,29,46,147)(12,146,47,28)(13,27,48,145)(14,144,49,26)(15,25,50,143)(16,142,41,24)(17,23,42,141)(18,150,43,22)(19,21,44,149)(20,148,45,30)(51,83,96,116)(52,115,97,82)(53,81,98,114)(54,113,99,90)(55,89,100,112)(56,111,91,88)(57,87,92,120)(58,119,93,86)(59,85,94,118)(60,117,95,84)(61,109,76,131)(62,140,77,108)(63,107,78,139)(64,138,79,106)(65,105,80,137)(66,136,71,104)(67,103,72,135)(68,134,73,102)(69,101,74,133)(70,132,75,110) );
G=PermutationGroup([[(1,108,33,140),(2,114,34,81),(3,110,35,132),(4,116,36,83),(5,102,37,134),(6,118,38,85),(7,104,39,136),(8,120,40,87),(9,106,31,138),(10,112,32,89),(11,131,46,109),(12,82,47,115),(13,133,48,101),(14,84,49,117),(15,135,50,103),(16,86,41,119),(17,137,42,105),(18,88,43,111),(19,139,44,107),(20,90,45,113),(21,100,149,55),(22,64,150,79),(23,92,141,57),(24,66,142,71),(25,94,143,59),(26,68,144,73),(27,96,145,51),(28,70,146,75),(29,98,147,53),(30,62,148,77),(52,123,97,155),(54,125,99,157),(56,127,91,159),(58,129,93,151),(60,121,95,153),(61,156,76,124),(63,158,78,126),(65,160,80,128),(67,152,72,130),(69,154,74,122)], [(1,95,20,73),(2,51,11,69),(3,97,12,75),(4,53,13,61),(5,99,14,77),(6,55,15,63),(7,91,16,79),(8,57,17,65),(9,93,18,71),(10,59,19,67),(21,135,158,118),(22,104,159,86),(23,137,160,120),(24,106,151,88),(25,139,152,112),(26,108,153,90),(27,131,154,114),(28,110,155,82),(29,133,156,116),(30,102,157,84),(31,58,43,66),(32,94,44,72),(33,60,45,68),(34,96,46,74),(35,52,47,70),(36,98,48,76),(37,54,49,62),(38,100,50,78),(39,56,41,64),(40,92,42,80),(81,145,109,122),(83,147,101,124),(85,149,103,126),(87,141,105,128),(89,143,107,130),(111,142,138,129),(113,144,140,121),(115,146,132,123),(117,148,134,125),(119,150,136,127)], [(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)], [(1,125,33,157),(2,156,34,124),(3,123,35,155),(4,154,36,122),(5,121,37,153),(6,152,38,130),(7,129,39,151),(8,160,40,128),(9,127,31,159),(10,158,32,126),(11,29,46,147),(12,146,47,28),(13,27,48,145),(14,144,49,26),(15,25,50,143),(16,142,41,24),(17,23,42,141),(18,150,43,22),(19,21,44,149),(20,148,45,30),(51,83,96,116),(52,115,97,82),(53,81,98,114),(54,113,99,90),(55,89,100,112),(56,111,91,88),(57,87,92,120),(58,119,93,86),(59,85,94,118),(60,117,95,84),(61,109,76,131),(62,140,77,108),(63,107,78,139),(64,138,79,106),(65,105,80,137),(66,136,71,104),(67,103,72,135),(68,134,73,102),(69,101,74,133),(70,132,75,110)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D5×C4○D4 |
| kernel | C42.189D10 | D5×C42 | C42⋊2D5 | C23.11D10 | Dic5⋊4D4 | D10.12D4 | D10⋊D4 | Dic5.5D4 | Dic5⋊3Q8 | Dic5.Q8 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | D10⋊Q8 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | D10 | C42 | C22⋊C4 | C4⋊C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 4 | 2 | 6 | 6 | 12 |
Matrix representation of C42.189D10 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 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 | 0 | 9 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 32 | 40 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 0 | 32 | 40 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,39,9],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,35,0,0,0,0,6,6,0,0,0,0,0,0,1,32,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,6,0,0,0,0,0,1,0,0,0,0,0,0,1,32,0,0,0,0,23,40] >;
C42.189D10 in GAP, Magma, Sage, TeX
C_4^2._{189}D_{10} % in TeX
G:=Group("C4^2.189D10"); // GroupNames label
G:=SmallGroup(320,1378);
// by ID
G=gap.SmallGroup(320,1378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,794,297,12550]);
// 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^-1*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations