metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.62D10, (C2×C20).81D4, C20⋊2Q8⋊17C2, (C2×D4).45D10, (C2×Q8).35D10, C4.4D4.5D5, C20.66(C4○D4), Q8⋊Dic5⋊20C2, C42.D5⋊8C2, C4.20(D4⋊2D5), (C2×C20).373C23, (C4×C20).104C22, D4⋊Dic5.12C2, (D4×C10).61C22, (Q8×C10).53C22, C2.17(D4⋊D10), C10.118(C8⋊C22), C10.41(C4.4D4), C2.8(C20.17D4), C4⋊Dic5.150C22, C2.18(D4.9D10), C10.119(C8.C22), C5⋊4(C42.28C22), (C2×C10).504(C2×D4), (C2×C4).60(C5⋊D4), (C5×C4.4D4).3C2, (C2×C4).473(C22×D5), C22.179(C2×C5⋊D4), (C2×C5⋊2C8).120C22, SmallGroup(320,682)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.62D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 350 in 100 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.28C22, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C22⋊C4, C2×Dic10, D4×C10, Q8×C10, C42.D5, D4⋊Dic5, Q8⋊Dic5, C20⋊2Q8, C5×C4.4D4, C42.62D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C42.28C22, D4⋊2D5, C2×C5⋊D4, C20.17D4, D4⋊D10, D4.9D10, C42.62D10
►On 160 points
Generators in S160
(1 42 6 57)(2 48 7 53)(3 44 8 59)(4 50 9 55)(5 46 10 51)(11 52 20 47)(12 58 16 43)(13 54 17 49)(14 60 18 45)(15 56 19 41)(21 88 127 98)(22 111 128 101)(23 90 129 100)(24 113 130 103)(25 82 121 92)(26 115 122 105)(27 84 123 94)(28 117 124 107)(29 86 125 96)(30 119 126 109)(31 70 146 65)(32 77 147 72)(33 62 148 67)(34 79 149 74)(35 64 150 69)(36 71 141 76)(37 66 142 61)(38 73 143 78)(39 68 144 63)(40 75 145 80)(81 157 91 140)(83 159 93 132)(85 151 95 134)(87 153 97 136)(89 155 99 138)(102 139 112 156)(104 131 114 158)(106 133 116 160)(108 135 118 152)(110 137 120 154)
(1 35 11 40)(2 31 12 36)(3 37 13 32)(4 33 14 38)(5 39 15 34)(6 150 20 145)(7 146 16 141)(8 142 17 147)(9 148 18 143)(10 144 19 149)(21 26 154 159)(22 160 155 27)(23 28 156 151)(24 152 157 29)(25 30 158 153)(41 74 51 63)(42 64 52 75)(43 76 53 65)(44 66 54 77)(45 78 55 67)(46 68 56 79)(47 80 57 69)(48 70 58 71)(49 72 59 61)(50 62 60 73)(81 96 103 118)(82 119 104 97)(83 98 105 120)(84 111 106 99)(85 100 107 112)(86 113 108 91)(87 92 109 114)(88 115 110 93)(89 94 101 116)(90 117 102 95)(121 126 131 136)(122 137 132 127)(123 128 133 138)(124 139 134 129)(125 130 135 140)
(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 97 6 87)(2 91 7 81)(3 95 8 85)(4 99 9 89)(5 93 10 83)(11 119 20 109)(12 113 16 103)(13 117 17 107)(14 111 18 101)(15 115 19 105)(21 74 127 79)(22 45 128 60)(23 72 129 77)(24 43 130 58)(25 80 121 75)(26 41 122 56)(27 78 123 73)(28 49 124 54)(29 76 125 71)(30 47 126 52)(31 108 146 118)(32 90 147 100)(33 106 148 116)(34 88 149 98)(35 104 150 114)(36 86 141 96)(37 102 142 112)(38 84 143 94)(39 110 144 120)(40 82 145 92)(42 153 57 136)(44 151 59 134)(46 159 51 132)(48 157 53 140)(50 155 55 138)(61 139 66 156)(62 160 67 133)(63 137 68 154)(64 158 69 131)(65 135 70 152)
G:=sub<Sym(160)| (1,42,6,57)(2,48,7,53)(3,44,8,59)(4,50,9,55)(5,46,10,51)(11,52,20,47)(12,58,16,43)(13,54,17,49)(14,60,18,45)(15,56,19,41)(21,88,127,98)(22,111,128,101)(23,90,129,100)(24,113,130,103)(25,82,121,92)(26,115,122,105)(27,84,123,94)(28,117,124,107)(29,86,125,96)(30,119,126,109)(31,70,146,65)(32,77,147,72)(33,62,148,67)(34,79,149,74)(35,64,150,69)(36,71,141,76)(37,66,142,61)(38,73,143,78)(39,68,144,63)(40,75,145,80)(81,157,91,140)(83,159,93,132)(85,151,95,134)(87,153,97,136)(89,155,99,138)(102,139,112,156)(104,131,114,158)(106,133,116,160)(108,135,118,152)(110,137,120,154), (1,35,11,40)(2,31,12,36)(3,37,13,32)(4,33,14,38)(5,39,15,34)(6,150,20,145)(7,146,16,141)(8,142,17,147)(9,148,18,143)(10,144,19,149)(21,26,154,159)(22,160,155,27)(23,28,156,151)(24,152,157,29)(25,30,158,153)(41,74,51,63)(42,64,52,75)(43,76,53,65)(44,66,54,77)(45,78,55,67)(46,68,56,79)(47,80,57,69)(48,70,58,71)(49,72,59,61)(50,62,60,73)(81,96,103,118)(82,119,104,97)(83,98,105,120)(84,111,106,99)(85,100,107,112)(86,113,108,91)(87,92,109,114)(88,115,110,93)(89,94,101,116)(90,117,102,95)(121,126,131,136)(122,137,132,127)(123,128,133,138)(124,139,134,129)(125,130,135,140), (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), (1,97,6,87)(2,91,7,81)(3,95,8,85)(4,99,9,89)(5,93,10,83)(11,119,20,109)(12,113,16,103)(13,117,17,107)(14,111,18,101)(15,115,19,105)(21,74,127,79)(22,45,128,60)(23,72,129,77)(24,43,130,58)(25,80,121,75)(26,41,122,56)(27,78,123,73)(28,49,124,54)(29,76,125,71)(30,47,126,52)(31,108,146,118)(32,90,147,100)(33,106,148,116)(34,88,149,98)(35,104,150,114)(36,86,141,96)(37,102,142,112)(38,84,143,94)(39,110,144,120)(40,82,145,92)(42,153,57,136)(44,151,59,134)(46,159,51,132)(48,157,53,140)(50,155,55,138)(61,139,66,156)(62,160,67,133)(63,137,68,154)(64,158,69,131)(65,135,70,152)>;
G:=Group( (1,42,6,57)(2,48,7,53)(3,44,8,59)(4,50,9,55)(5,46,10,51)(11,52,20,47)(12,58,16,43)(13,54,17,49)(14,60,18,45)(15,56,19,41)(21,88,127,98)(22,111,128,101)(23,90,129,100)(24,113,130,103)(25,82,121,92)(26,115,122,105)(27,84,123,94)(28,117,124,107)(29,86,125,96)(30,119,126,109)(31,70,146,65)(32,77,147,72)(33,62,148,67)(34,79,149,74)(35,64,150,69)(36,71,141,76)(37,66,142,61)(38,73,143,78)(39,68,144,63)(40,75,145,80)(81,157,91,140)(83,159,93,132)(85,151,95,134)(87,153,97,136)(89,155,99,138)(102,139,112,156)(104,131,114,158)(106,133,116,160)(108,135,118,152)(110,137,120,154), (1,35,11,40)(2,31,12,36)(3,37,13,32)(4,33,14,38)(5,39,15,34)(6,150,20,145)(7,146,16,141)(8,142,17,147)(9,148,18,143)(10,144,19,149)(21,26,154,159)(22,160,155,27)(23,28,156,151)(24,152,157,29)(25,30,158,153)(41,74,51,63)(42,64,52,75)(43,76,53,65)(44,66,54,77)(45,78,55,67)(46,68,56,79)(47,80,57,69)(48,70,58,71)(49,72,59,61)(50,62,60,73)(81,96,103,118)(82,119,104,97)(83,98,105,120)(84,111,106,99)(85,100,107,112)(86,113,108,91)(87,92,109,114)(88,115,110,93)(89,94,101,116)(90,117,102,95)(121,126,131,136)(122,137,132,127)(123,128,133,138)(124,139,134,129)(125,130,135,140), (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), (1,97,6,87)(2,91,7,81)(3,95,8,85)(4,99,9,89)(5,93,10,83)(11,119,20,109)(12,113,16,103)(13,117,17,107)(14,111,18,101)(15,115,19,105)(21,74,127,79)(22,45,128,60)(23,72,129,77)(24,43,130,58)(25,80,121,75)(26,41,122,56)(27,78,123,73)(28,49,124,54)(29,76,125,71)(30,47,126,52)(31,108,146,118)(32,90,147,100)(33,106,148,116)(34,88,149,98)(35,104,150,114)(36,86,141,96)(37,102,142,112)(38,84,143,94)(39,110,144,120)(40,82,145,92)(42,153,57,136)(44,151,59,134)(46,159,51,132)(48,157,53,140)(50,155,55,138)(61,139,66,156)(62,160,67,133)(63,137,68,154)(64,158,69,131)(65,135,70,152) );
G=PermutationGroup([[(1,42,6,57),(2,48,7,53),(3,44,8,59),(4,50,9,55),(5,46,10,51),(11,52,20,47),(12,58,16,43),(13,54,17,49),(14,60,18,45),(15,56,19,41),(21,88,127,98),(22,111,128,101),(23,90,129,100),(24,113,130,103),(25,82,121,92),(26,115,122,105),(27,84,123,94),(28,117,124,107),(29,86,125,96),(30,119,126,109),(31,70,146,65),(32,77,147,72),(33,62,148,67),(34,79,149,74),(35,64,150,69),(36,71,141,76),(37,66,142,61),(38,73,143,78),(39,68,144,63),(40,75,145,80),(81,157,91,140),(83,159,93,132),(85,151,95,134),(87,153,97,136),(89,155,99,138),(102,139,112,156),(104,131,114,158),(106,133,116,160),(108,135,118,152),(110,137,120,154)], [(1,35,11,40),(2,31,12,36),(3,37,13,32),(4,33,14,38),(5,39,15,34),(6,150,20,145),(7,146,16,141),(8,142,17,147),(9,148,18,143),(10,144,19,149),(21,26,154,159),(22,160,155,27),(23,28,156,151),(24,152,157,29),(25,30,158,153),(41,74,51,63),(42,64,52,75),(43,76,53,65),(44,66,54,77),(45,78,55,67),(46,68,56,79),(47,80,57,69),(48,70,58,71),(49,72,59,61),(50,62,60,73),(81,96,103,118),(82,119,104,97),(83,98,105,120),(84,111,106,99),(85,100,107,112),(86,113,108,91),(87,92,109,114),(88,115,110,93),(89,94,101,116),(90,117,102,95),(121,126,131,136),(122,137,132,127),(123,128,133,138),(124,139,134,129),(125,130,135,140)], [(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)], [(1,97,6,87),(2,91,7,81),(3,95,8,85),(4,99,9,89),(5,93,10,83),(11,119,20,109),(12,113,16,103),(13,117,17,107),(14,111,18,101),(15,115,19,105),(21,74,127,79),(22,45,128,60),(23,72,129,77),(24,43,130,58),(25,80,121,75),(26,41,122,56),(27,78,123,73),(28,49,124,54),(29,76,125,71),(30,47,126,52),(31,108,146,118),(32,90,147,100),(33,106,148,116),(34,88,149,98),(35,104,150,114),(36,86,141,96),(37,102,142,112),(38,84,143,94),(39,110,144,120),(40,82,145,92),(42,153,57,136),(44,151,59,134),(46,159,51,132),(48,157,53,140),(50,155,55,138),(61,139,66,156),(62,160,67,133),(63,137,68,154),(64,158,69,131),(65,135,70,152)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 8 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C8⋊C22 | C8.C22 | D4⋊2D5 | D4⋊D10 | D4.9D10 |
| kernel | C42.62D10 | C42.D5 | D4⋊Dic5 | Q8⋊Dic5 | C20⋊2Q8 | C5×C4.4D4 | C2×C20 | C4.4D4 | C20 | C42 | C2×D4 | C2×Q8 | C2×C4 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.62D10 ►in GL8(𝔽41)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 0 | 16 | 32 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 27 | 13 | 22 |
| 0 | 0 | 0 | 0 | 0 | 40 | 28 | 13 |
| 0 | 0 | 0 | 0 | 24 | 21 | 1 | 14 |
| 0 | 0 | 0 | 0 | 24 | 24 | 0 | 15 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 33 | 34 | 34 |
| 0 | 0 | 0 | 0 | 40 | 36 | 7 | 1 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 30 | 31 | 10 |
| 0 | 0 | 0 | 0 | 35 | 5 | 10 | 39 |
| 0 | 0 | 0 | 0 | 40 | 36 | 30 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 0 | 6 |
G:=sub<GL(8,GF(41))| [0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30,32,25,0,0,0,0,0,9,11,0,16,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,0,24,24,0,0,0,0,27,40,21,24,0,0,0,0,13,28,1,0,0,0,0,0,22,13,14,15],[7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,40,34,36,40,0,0,0,0,7,7,33,36,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1],[0,0,32,22,0,0,0,0,0,0,0,9,0,0,0,0,32,22,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,35,40,21,0,0,0,0,30,5,36,21,0,0,0,0,31,10,30,0,0,0,0,0,10,39,0,6] >;
C42.62D10 in GAP, Magma, Sage, TeX
C_4^2._{62}D_{10} % in TeX
G:=Group("C4^2.62D10"); // GroupNames label
G:=SmallGroup(320,682);
// by ID
G=gap.SmallGroup(320,682);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,590,471,438,102,12550]);
// 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=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations