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, D10.19(C4○D4), Dic5⋊3Q8⋊41C2, (C2×C10).250C24, (C4×C20).234C22, D10.12D4⋊50C2, D10.13D4⋊40C2, C23.56(C22×D5), Dic5.21(C4○D4), Dic5.5D4⋊46C2, Dic5.Q8⋊37C2, (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
Subgroups: 798 in 234 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×14], C22, C22 [×10], C5, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×3], C10 [×3], C10, C42, C42 [×5], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×4], Dic5 [×4], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2, C42⋊2C2, Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×6], C22×D5 [×2], C22×C10, C23.36C23, C4×Dic5 [×5], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×6], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], D5×C42, C42⋊2D5, C23.11D10, Dic5⋊4D4 [×2], 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 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C23×D5, D5×C4○D4 [×3], C42.189D10
Generators and relations
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 >
►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)(11 135 50 103)(12 86 41 119)(13 137 42 105)(14 88 43 111)(15 139 44 107)(16 90 45 113)(17 131 46 109)(18 82 47 115)(19 133 48 101)(20 84 49 117)(21 100 141 55)(22 64 142 79)(23 92 143 57)(24 66 144 71)(25 94 145 59)(26 68 146 73)(27 96 147 51)(28 70 148 75)(29 98 149 53)(30 62 150 77)(52 125 97 155)(54 127 99 157)(56 129 91 159)(58 121 93 151)(60 123 95 153)(61 156 76 126)(63 158 78 128)(65 160 80 130)(67 152 72 122)(69 154 74 124)
(1 95 16 73)(2 51 17 69)(3 97 18 75)(4 53 19 61)(5 99 20 77)(6 55 11 63)(7 91 12 79)(8 57 13 65)(9 93 14 71)(10 59 15 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)(81 147 109 124)(83 149 101 126)(85 141 103 128)(87 143 105 130)(89 145 107 122)(111 144 138 121)(113 146 140 123)(115 148 132 125)(117 150 134 127)(119 142 136 129)
(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)
(1 127 33 157)(2 156 34 126)(3 125 35 155)(4 154 36 124)(5 123 37 153)(6 152 38 122)(7 121 39 151)(8 160 40 130)(9 129 31 159)(10 158 32 128)(11 25 50 145)(12 144 41 24)(13 23 42 143)(14 142 43 22)(15 21 44 141)(16 150 45 30)(17 29 46 149)(18 148 47 28)(19 27 48 147)(20 146 49 26)(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)(11,135,50,103)(12,86,41,119)(13,137,42,105)(14,88,43,111)(15,139,44,107)(16,90,45,113)(17,131,46,109)(18,82,47,115)(19,133,48,101)(20,84,49,117)(21,100,141,55)(22,64,142,79)(23,92,143,57)(24,66,144,71)(25,94,145,59)(26,68,146,73)(27,96,147,51)(28,70,148,75)(29,98,149,53)(30,62,150,77)(52,125,97,155)(54,127,99,157)(56,129,91,159)(58,121,93,151)(60,123,95,153)(61,156,76,126)(63,158,78,128)(65,160,80,130)(67,152,72,122)(69,154,74,124), (1,95,16,73)(2,51,17,69)(3,97,18,75)(4,53,19,61)(5,99,20,77)(6,55,11,63)(7,91,12,79)(8,57,13,65)(9,93,14,71)(10,59,15,67)(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)(81,147,109,124)(83,149,101,126)(85,141,103,128)(87,143,105,130)(89,145,107,122)(111,144,138,121)(113,146,140,123)(115,148,132,125)(117,150,134,127)(119,142,136,129), (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), (1,127,33,157)(2,156,34,126)(3,125,35,155)(4,154,36,124)(5,123,37,153)(6,152,38,122)(7,121,39,151)(8,160,40,130)(9,129,31,159)(10,158,32,128)(11,25,50,145)(12,144,41,24)(13,23,42,143)(14,142,43,22)(15,21,44,141)(16,150,45,30)(17,29,46,149)(18,148,47,28)(19,27,48,147)(20,146,49,26)(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)(11,135,50,103)(12,86,41,119)(13,137,42,105)(14,88,43,111)(15,139,44,107)(16,90,45,113)(17,131,46,109)(18,82,47,115)(19,133,48,101)(20,84,49,117)(21,100,141,55)(22,64,142,79)(23,92,143,57)(24,66,144,71)(25,94,145,59)(26,68,146,73)(27,96,147,51)(28,70,148,75)(29,98,149,53)(30,62,150,77)(52,125,97,155)(54,127,99,157)(56,129,91,159)(58,121,93,151)(60,123,95,153)(61,156,76,126)(63,158,78,128)(65,160,80,130)(67,152,72,122)(69,154,74,124), (1,95,16,73)(2,51,17,69)(3,97,18,75)(4,53,19,61)(5,99,20,77)(6,55,11,63)(7,91,12,79)(8,57,13,65)(9,93,14,71)(10,59,15,67)(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)(81,147,109,124)(83,149,101,126)(85,141,103,128)(87,143,105,130)(89,145,107,122)(111,144,138,121)(113,146,140,123)(115,148,132,125)(117,150,134,127)(119,142,136,129), (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), (1,127,33,157)(2,156,34,126)(3,125,35,155)(4,154,36,124)(5,123,37,153)(6,152,38,122)(7,121,39,151)(8,160,40,130)(9,129,31,159)(10,158,32,128)(11,25,50,145)(12,144,41,24)(13,23,42,143)(14,142,43,22)(15,21,44,141)(16,150,45,30)(17,29,46,149)(18,148,47,28)(19,27,48,147)(20,146,49,26)(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),(11,135,50,103),(12,86,41,119),(13,137,42,105),(14,88,43,111),(15,139,44,107),(16,90,45,113),(17,131,46,109),(18,82,47,115),(19,133,48,101),(20,84,49,117),(21,100,141,55),(22,64,142,79),(23,92,143,57),(24,66,144,71),(25,94,145,59),(26,68,146,73),(27,96,147,51),(28,70,148,75),(29,98,149,53),(30,62,150,77),(52,125,97,155),(54,127,99,157),(56,129,91,159),(58,121,93,151),(60,123,95,153),(61,156,76,126),(63,158,78,128),(65,160,80,130),(67,152,72,122),(69,154,74,124)], [(1,95,16,73),(2,51,17,69),(3,97,18,75),(4,53,19,61),(5,99,20,77),(6,55,11,63),(7,91,12,79),(8,57,13,65),(9,93,14,71),(10,59,15,67),(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),(81,147,109,124),(83,149,101,126),(85,141,103,128),(87,143,105,130),(89,145,107,122),(111,144,138,121),(113,146,140,123),(115,148,132,125),(117,150,134,127),(119,142,136,129)], [(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)], [(1,127,33,157),(2,156,34,126),(3,125,35,155),(4,154,36,124),(5,123,37,153),(6,152,38,122),(7,121,39,151),(8,160,40,130),(9,129,31,159),(10,158,32,128),(11,25,50,145),(12,144,41,24),(13,23,42,143),(14,142,43,22),(15,21,44,141),(16,150,45,30),(17,29,46,149),(18,148,47,28),(19,27,48,147),(20,146,49,26),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀