metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.165D10, C10.1432+ 1+4, C10.1032- 1+4, C20⋊Q8⋊42C2, C20⋊2Q8⋊9C2, C4⋊C4.120D10, C42⋊2C2⋊9D5, C42⋊2D5⋊2C2, (C4×C20).9C22, D10⋊Q8⋊46C2, D10⋊2Q8⋊43C2, (C2×C20).97C23, C22⋊C4.42D10, C4.Dic10⋊41C2, (C2×C10).256C24, C4⋊Dic5.55C22, C2.68(D4⋊8D10), C23.62(C22×D5), Dic5.Q8⋊40C2, D10.12D4.5C2, C23.D10⋊47C2, (C22×C10).70C23, Dic5.5D4.5C2, C22.D20.4C2, C22.277(C23×D5), Dic5.14D4⋊48C2, C23.D5.70C22, D10⋊C4.48C22, C5⋊5(C22.57C24), (C2×Dic5).132C23, (C2×Dic10).45C22, (C4×Dic5).161C22, C10.D4.11C22, (C22×D5).115C23, C2.67(D4.10D10), (C22×Dic5).155C22, C4⋊C4⋊D5⋊45C2, (C2×C4×D5).146C22, (C5×C42⋊2C2)⋊11C2, (C5×C4⋊C4).207C22, (C2×C4).212(C22×D5), (C2×C5⋊D4).76C22, (C5×C22⋊C4).81C22, SmallGroup(320,1384)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.165D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
Subgroups: 678 in 196 conjugacy classes, 91 normal (all 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, C20, D10, C2×C10, C2×C10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C22.57C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C20⋊2Q8, C42⋊2D5, Dic5.14D4, C23.D10, D10.12D4, Dic5.5D4, C22.D20, C20⋊Q8, Dic5.Q8, C4.Dic10, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.165D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.57C24, C23×D5, D4⋊8D10, D4.10D10, C42.165D10
►On 160 points
Generators in S160
(1 102 57 31)(2 22 58 113)(3 104 59 33)(4 24 60 115)(5 106 41 35)(6 26 42 117)(7 108 43 37)(8 28 44 119)(9 110 45 39)(10 30 46 101)(11 112 47 21)(12 32 48 103)(13 114 49 23)(14 34 50 105)(15 116 51 25)(16 36 52 107)(17 118 53 27)(18 38 54 109)(19 120 55 29)(20 40 56 111)(61 156 83 122)(62 133 84 147)(63 158 85 124)(64 135 86 149)(65 160 87 126)(66 137 88 151)(67 142 89 128)(68 139 90 153)(69 144 91 130)(70 121 92 155)(71 146 93 132)(72 123 94 157)(73 148 95 134)(74 125 96 159)(75 150 97 136)(76 127 98 141)(77 152 99 138)(78 129 100 143)(79 154 81 140)(80 131 82 145)
(1 81 47 69)(2 92 48 80)(3 83 49 71)(4 94 50 62)(5 85 51 73)(6 96 52 64)(7 87 53 75)(8 98 54 66)(9 89 55 77)(10 100 56 68)(11 91 57 79)(12 82 58 70)(13 93 59 61)(14 84 60 72)(15 95 41 63)(16 86 42 74)(17 97 43 65)(18 88 44 76)(19 99 45 67)(20 90 46 78)(21 144 102 140)(22 155 103 131)(23 146 104 122)(24 157 105 133)(25 148 106 124)(26 159 107 135)(27 150 108 126)(28 141 109 137)(29 152 110 128)(30 143 111 139)(31 154 112 130)(32 145 113 121)(33 156 114 132)(34 147 115 123)(35 158 116 134)(36 149 117 125)(37 160 118 136)(38 151 119 127)(39 142 120 138)(40 153 101 129)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 40 31 30)(22 29 32 39)(23 38 33 28)(24 27 34 37)(25 36 35 26)(41 42 51 52)(43 60 53 50)(44 49 54 59)(45 58 55 48)(46 47 56 57)(61 88 71 98)(62 97 72 87)(63 86 73 96)(64 95 74 85)(65 84 75 94)(66 93 76 83)(67 82 77 92)(68 91 78 81)(69 100 79 90)(70 89 80 99)(101 112 111 102)(103 110 113 120)(104 119 114 109)(105 108 115 118)(106 117 116 107)(121 142 131 152)(122 151 132 141)(123 160 133 150)(124 149 134 159)(125 158 135 148)(126 147 136 157)(127 156 137 146)(128 145 138 155)(129 154 139 144)(130 143 140 153)
G:=sub<Sym(160)| (1,102,57,31)(2,22,58,113)(3,104,59,33)(4,24,60,115)(5,106,41,35)(6,26,42,117)(7,108,43,37)(8,28,44,119)(9,110,45,39)(10,30,46,101)(11,112,47,21)(12,32,48,103)(13,114,49,23)(14,34,50,105)(15,116,51,25)(16,36,52,107)(17,118,53,27)(18,38,54,109)(19,120,55,29)(20,40,56,111)(61,156,83,122)(62,133,84,147)(63,158,85,124)(64,135,86,149)(65,160,87,126)(66,137,88,151)(67,142,89,128)(68,139,90,153)(69,144,91,130)(70,121,92,155)(71,146,93,132)(72,123,94,157)(73,148,95,134)(74,125,96,159)(75,150,97,136)(76,127,98,141)(77,152,99,138)(78,129,100,143)(79,154,81,140)(80,131,82,145), (1,81,47,69)(2,92,48,80)(3,83,49,71)(4,94,50,62)(5,85,51,73)(6,96,52,64)(7,87,53,75)(8,98,54,66)(9,89,55,77)(10,100,56,68)(11,91,57,79)(12,82,58,70)(13,93,59,61)(14,84,60,72)(15,95,41,63)(16,86,42,74)(17,97,43,65)(18,88,44,76)(19,99,45,67)(20,90,46,78)(21,144,102,140)(22,155,103,131)(23,146,104,122)(24,157,105,133)(25,148,106,124)(26,159,107,135)(27,150,108,126)(28,141,109,137)(29,152,110,128)(30,143,111,139)(31,154,112,130)(32,145,113,121)(33,156,114,132)(34,147,115,123)(35,158,116,134)(36,149,117,125)(37,160,118,136)(38,151,119,127)(39,142,120,138)(40,153,101,129), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,40,31,30)(22,29,32,39)(23,38,33,28)(24,27,34,37)(25,36,35,26)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)(61,88,71,98)(62,97,72,87)(63,86,73,96)(64,95,74,85)(65,84,75,94)(66,93,76,83)(67,82,77,92)(68,91,78,81)(69,100,79,90)(70,89,80,99)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)(121,142,131,152)(122,151,132,141)(123,160,133,150)(124,149,134,159)(125,158,135,148)(126,147,136,157)(127,156,137,146)(128,145,138,155)(129,154,139,144)(130,143,140,153)>;
G:=Group( (1,102,57,31)(2,22,58,113)(3,104,59,33)(4,24,60,115)(5,106,41,35)(6,26,42,117)(7,108,43,37)(8,28,44,119)(9,110,45,39)(10,30,46,101)(11,112,47,21)(12,32,48,103)(13,114,49,23)(14,34,50,105)(15,116,51,25)(16,36,52,107)(17,118,53,27)(18,38,54,109)(19,120,55,29)(20,40,56,111)(61,156,83,122)(62,133,84,147)(63,158,85,124)(64,135,86,149)(65,160,87,126)(66,137,88,151)(67,142,89,128)(68,139,90,153)(69,144,91,130)(70,121,92,155)(71,146,93,132)(72,123,94,157)(73,148,95,134)(74,125,96,159)(75,150,97,136)(76,127,98,141)(77,152,99,138)(78,129,100,143)(79,154,81,140)(80,131,82,145), (1,81,47,69)(2,92,48,80)(3,83,49,71)(4,94,50,62)(5,85,51,73)(6,96,52,64)(7,87,53,75)(8,98,54,66)(9,89,55,77)(10,100,56,68)(11,91,57,79)(12,82,58,70)(13,93,59,61)(14,84,60,72)(15,95,41,63)(16,86,42,74)(17,97,43,65)(18,88,44,76)(19,99,45,67)(20,90,46,78)(21,144,102,140)(22,155,103,131)(23,146,104,122)(24,157,105,133)(25,148,106,124)(26,159,107,135)(27,150,108,126)(28,141,109,137)(29,152,110,128)(30,143,111,139)(31,154,112,130)(32,145,113,121)(33,156,114,132)(34,147,115,123)(35,158,116,134)(36,149,117,125)(37,160,118,136)(38,151,119,127)(39,142,120,138)(40,153,101,129), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,40,31,30)(22,29,32,39)(23,38,33,28)(24,27,34,37)(25,36,35,26)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)(61,88,71,98)(62,97,72,87)(63,86,73,96)(64,95,74,85)(65,84,75,94)(66,93,76,83)(67,82,77,92)(68,91,78,81)(69,100,79,90)(70,89,80,99)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)(121,142,131,152)(122,151,132,141)(123,160,133,150)(124,149,134,159)(125,158,135,148)(126,147,136,157)(127,156,137,146)(128,145,138,155)(129,154,139,144)(130,143,140,153) );
G=PermutationGroup([[(1,102,57,31),(2,22,58,113),(3,104,59,33),(4,24,60,115),(5,106,41,35),(6,26,42,117),(7,108,43,37),(8,28,44,119),(9,110,45,39),(10,30,46,101),(11,112,47,21),(12,32,48,103),(13,114,49,23),(14,34,50,105),(15,116,51,25),(16,36,52,107),(17,118,53,27),(18,38,54,109),(19,120,55,29),(20,40,56,111),(61,156,83,122),(62,133,84,147),(63,158,85,124),(64,135,86,149),(65,160,87,126),(66,137,88,151),(67,142,89,128),(68,139,90,153),(69,144,91,130),(70,121,92,155),(71,146,93,132),(72,123,94,157),(73,148,95,134),(74,125,96,159),(75,150,97,136),(76,127,98,141),(77,152,99,138),(78,129,100,143),(79,154,81,140),(80,131,82,145)], [(1,81,47,69),(2,92,48,80),(3,83,49,71),(4,94,50,62),(5,85,51,73),(6,96,52,64),(7,87,53,75),(8,98,54,66),(9,89,55,77),(10,100,56,68),(11,91,57,79),(12,82,58,70),(13,93,59,61),(14,84,60,72),(15,95,41,63),(16,86,42,74),(17,97,43,65),(18,88,44,76),(19,99,45,67),(20,90,46,78),(21,144,102,140),(22,155,103,131),(23,146,104,122),(24,157,105,133),(25,148,106,124),(26,159,107,135),(27,150,108,126),(28,141,109,137),(29,152,110,128),(30,143,111,139),(31,154,112,130),(32,145,113,121),(33,156,114,132),(34,147,115,123),(35,158,116,134),(36,149,117,125),(37,160,118,136),(38,151,119,127),(39,142,120,138),(40,153,101,129)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,40,31,30),(22,29,32,39),(23,38,33,28),(24,27,34,37),(25,36,35,26),(41,42,51,52),(43,60,53,50),(44,49,54,59),(45,58,55,48),(46,47,56,57),(61,88,71,98),(62,97,72,87),(63,86,73,96),(64,95,74,85),(65,84,75,94),(66,93,76,83),(67,82,77,92),(68,91,78,81),(69,100,79,90),(70,89,80,99),(101,112,111,102),(103,110,113,120),(104,119,114,109),(105,108,115,118),(106,117,116,107),(121,142,131,152),(122,151,132,141),(123,160,133,150),(124,149,134,159),(125,158,135,148),(126,147,136,157),(127,156,137,146),(128,145,138,155),(129,154,139,144),(130,143,140,153)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.165D10 | C20⋊2Q8 | C42⋊2D5 | Dic5.14D4 | C23.D10 | D10.12D4 | Dic5.5D4 | C22.D20 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 4 | 8 |
Matrix representation of C42.165D10 ►in GL8(𝔽41)
| 2 | 13 | 37 | 15 | 0 | 0 | 0 | 0 |
| 28 | 39 | 26 | 4 | 0 | 0 | 0 | 0 |
| 2 | 13 | 39 | 28 | 0 | 0 | 0 | 0 |
| 28 | 39 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 2 |
| 1 | 0 | 39 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 39 | 0 | 0 | 0 | 0 |
| 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | 3 | 39 | 39 | 0 | 0 | 0 | 0 |
| 38 | 21 | 2 | 27 | 0 | 0 | 0 | 0 |
| 2 | 2 | 38 | 38 | 0 | 0 | 0 | 0 |
| 39 | 14 | 3 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 32 |
| 3 | 3 | 39 | 39 | 0 | 0 | 0 | 0 |
| 21 | 38 | 27 | 2 | 0 | 0 | 0 | 0 |
| 2 | 2 | 38 | 38 | 0 | 0 | 0 | 0 |
| 14 | 39 | 20 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 28 |
G:=sub<GL(8,GF(41))| [2,28,2,28,0,0,0,0,13,39,13,39,0,0,0,0,37,26,39,13,0,0,0,0,15,4,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,39,0,40,0,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,38,2,39,0,0,0,0,3,21,2,14,0,0,0,0,39,2,38,3,0,0,0,0,39,27,38,20,0,0,0,0,0,0,0,0,13,28,0,0,0,0,0,0,13,9,0,0,0,0,0,0,0,0,28,13,0,0,0,0,0,0,28,32],[3,21,2,14,0,0,0,0,3,38,2,39,0,0,0,0,39,27,38,20,0,0,0,0,39,2,38,3,0,0,0,0,0,0,0,0,13,9,0,0,0,0,0,0,13,28,0,0,0,0,0,0,0,0,13,9,0,0,0,0,0,0,13,28] >;
C42.165D10 in GAP, Magma, Sage, TeX
C_4^2._{165}D_{10} % in TeX
G:=Group("C4^2.165D10"); // GroupNames label
G:=SmallGroup(320,1384);
// by ID
G=gap.SmallGroup(320,1384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations