metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.156D10, C10.1342+ 1+4, C20⋊4D4⋊15C2, C4⋊D20⋊34C2, C4⋊C4.113D10, C42.C2⋊12D5, D20⋊8C4⋊38C2, C42⋊D5⋊22C2, C20.132(C4○D4), (C2×C20).189C23, (C4×C20).201C22, (C2×C10).242C24, C4.21(Q8⋊2D5), D10.13D4⋊36C2, C2.59(D4⋊8D10), (C2×D20).172C22, C22.263(C23×D5), C5⋊6(C22.34C24), (C4×Dic5).155C22, (C2×Dic5).272C23, (C22×D5).107C23, D10⋊C4.113C22, C10.D4.125C22, C10.119(C2×C4○D4), (C5×C42.C2)⋊15C2, C2.26(C2×Q8⋊2D5), (C2×C4×D5).141C22, (C5×C4⋊C4).197C22, (C2×C4).594(C22×D5), SmallGroup(320,1370)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.156D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 1070 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22.34C24, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C42⋊D5, C20⋊4D4, D20⋊8C4, D10.13D4, C4⋊D20, C5×C42.C2, C42.156D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.34C24, Q8⋊2D5, C23×D5, C2×Q8⋊2D5, D4⋊8D10, C42.156D10
►On 160 points
Generators in S160
(1 76 11 66)(2 147 12 157)(3 78 13 68)(4 149 14 159)(5 80 15 70)(6 151 16 141)(7 62 17 72)(8 153 18 143)(9 64 19 74)(10 155 20 145)(21 98 31 88)(22 45 32 55)(23 100 33 90)(24 47 34 57)(25 82 35 92)(26 49 36 59)(27 84 37 94)(28 51 38 41)(29 86 39 96)(30 53 40 43)(42 135 52 125)(44 137 54 127)(46 139 56 129)(48 121 58 131)(50 123 60 133)(61 108 71 118)(63 110 73 120)(65 112 75 102)(67 114 77 104)(69 116 79 106)(81 140 91 130)(83 122 93 132)(85 124 95 134)(87 126 97 136)(89 128 99 138)(101 154 111 144)(103 156 113 146)(105 158 115 148)(107 160 117 150)(109 142 119 152)
(1 52 113 86)(2 43 114 97)(3 54 115 88)(4 45 116 99)(5 56 117 90)(6 47 118 81)(7 58 119 92)(8 49 120 83)(9 60 101 94)(10 51 102 85)(11 42 103 96)(12 53 104 87)(13 44 105 98)(14 55 106 89)(15 46 107 100)(16 57 108 91)(17 48 109 82)(18 59 110 93)(19 50 111 84)(20 41 112 95)(21 78 127 148)(22 69 128 159)(23 80 129 150)(24 71 130 141)(25 62 131 152)(26 73 132 143)(27 64 133 154)(28 75 134 145)(29 66 135 156)(30 77 136 147)(31 68 137 158)(32 79 138 149)(33 70 139 160)(34 61 140 151)(35 72 121 142)(36 63 122 153)(37 74 123 144)(38 65 124 155)(39 76 125 146)(40 67 126 157)
(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 75 11 65)(2 64 12 74)(3 73 13 63)(4 62 14 72)(5 71 15 61)(6 80 16 70)(7 69 17 79)(8 78 18 68)(9 67 19 77)(10 76 20 66)(21 49 31 59)(22 58 32 48)(23 47 33 57)(24 56 34 46)(25 45 35 55)(26 54 36 44)(27 43 37 53)(28 52 38 42)(29 41 39 51)(30 50 40 60)(81 139 91 129)(82 128 92 138)(83 137 93 127)(84 126 94 136)(85 135 95 125)(86 124 96 134)(87 133 97 123)(88 122 98 132)(89 131 99 121)(90 140 100 130)(101 157 111 147)(102 146 112 156)(103 155 113 145)(104 144 114 154)(105 153 115 143)(106 142 116 152)(107 151 117 141)(108 160 118 150)(109 149 119 159)(110 158 120 148)
G:=sub<Sym(160)| (1,76,11,66)(2,147,12,157)(3,78,13,68)(4,149,14,159)(5,80,15,70)(6,151,16,141)(7,62,17,72)(8,153,18,143)(9,64,19,74)(10,155,20,145)(21,98,31,88)(22,45,32,55)(23,100,33,90)(24,47,34,57)(25,82,35,92)(26,49,36,59)(27,84,37,94)(28,51,38,41)(29,86,39,96)(30,53,40,43)(42,135,52,125)(44,137,54,127)(46,139,56,129)(48,121,58,131)(50,123,60,133)(61,108,71,118)(63,110,73,120)(65,112,75,102)(67,114,77,104)(69,116,79,106)(81,140,91,130)(83,122,93,132)(85,124,95,134)(87,126,97,136)(89,128,99,138)(101,154,111,144)(103,156,113,146)(105,158,115,148)(107,160,117,150)(109,142,119,152), (1,52,113,86)(2,43,114,97)(3,54,115,88)(4,45,116,99)(5,56,117,90)(6,47,118,81)(7,58,119,92)(8,49,120,83)(9,60,101,94)(10,51,102,85)(11,42,103,96)(12,53,104,87)(13,44,105,98)(14,55,106,89)(15,46,107,100)(16,57,108,91)(17,48,109,82)(18,59,110,93)(19,50,111,84)(20,41,112,95)(21,78,127,148)(22,69,128,159)(23,80,129,150)(24,71,130,141)(25,62,131,152)(26,73,132,143)(27,64,133,154)(28,75,134,145)(29,66,135,156)(30,77,136,147)(31,68,137,158)(32,79,138,149)(33,70,139,160)(34,61,140,151)(35,72,121,142)(36,63,122,153)(37,74,123,144)(38,65,124,155)(39,76,125,146)(40,67,126,157), (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), (1,75,11,65)(2,64,12,74)(3,73,13,63)(4,62,14,72)(5,71,15,61)(6,80,16,70)(7,69,17,79)(8,78,18,68)(9,67,19,77)(10,76,20,66)(21,49,31,59)(22,58,32,48)(23,47,33,57)(24,56,34,46)(25,45,35,55)(26,54,36,44)(27,43,37,53)(28,52,38,42)(29,41,39,51)(30,50,40,60)(81,139,91,129)(82,128,92,138)(83,137,93,127)(84,126,94,136)(85,135,95,125)(86,124,96,134)(87,133,97,123)(88,122,98,132)(89,131,99,121)(90,140,100,130)(101,157,111,147)(102,146,112,156)(103,155,113,145)(104,144,114,154)(105,153,115,143)(106,142,116,152)(107,151,117,141)(108,160,118,150)(109,149,119,159)(110,158,120,148)>;
G:=Group( (1,76,11,66)(2,147,12,157)(3,78,13,68)(4,149,14,159)(5,80,15,70)(6,151,16,141)(7,62,17,72)(8,153,18,143)(9,64,19,74)(10,155,20,145)(21,98,31,88)(22,45,32,55)(23,100,33,90)(24,47,34,57)(25,82,35,92)(26,49,36,59)(27,84,37,94)(28,51,38,41)(29,86,39,96)(30,53,40,43)(42,135,52,125)(44,137,54,127)(46,139,56,129)(48,121,58,131)(50,123,60,133)(61,108,71,118)(63,110,73,120)(65,112,75,102)(67,114,77,104)(69,116,79,106)(81,140,91,130)(83,122,93,132)(85,124,95,134)(87,126,97,136)(89,128,99,138)(101,154,111,144)(103,156,113,146)(105,158,115,148)(107,160,117,150)(109,142,119,152), (1,52,113,86)(2,43,114,97)(3,54,115,88)(4,45,116,99)(5,56,117,90)(6,47,118,81)(7,58,119,92)(8,49,120,83)(9,60,101,94)(10,51,102,85)(11,42,103,96)(12,53,104,87)(13,44,105,98)(14,55,106,89)(15,46,107,100)(16,57,108,91)(17,48,109,82)(18,59,110,93)(19,50,111,84)(20,41,112,95)(21,78,127,148)(22,69,128,159)(23,80,129,150)(24,71,130,141)(25,62,131,152)(26,73,132,143)(27,64,133,154)(28,75,134,145)(29,66,135,156)(30,77,136,147)(31,68,137,158)(32,79,138,149)(33,70,139,160)(34,61,140,151)(35,72,121,142)(36,63,122,153)(37,74,123,144)(38,65,124,155)(39,76,125,146)(40,67,126,157), (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), (1,75,11,65)(2,64,12,74)(3,73,13,63)(4,62,14,72)(5,71,15,61)(6,80,16,70)(7,69,17,79)(8,78,18,68)(9,67,19,77)(10,76,20,66)(21,49,31,59)(22,58,32,48)(23,47,33,57)(24,56,34,46)(25,45,35,55)(26,54,36,44)(27,43,37,53)(28,52,38,42)(29,41,39,51)(30,50,40,60)(81,139,91,129)(82,128,92,138)(83,137,93,127)(84,126,94,136)(85,135,95,125)(86,124,96,134)(87,133,97,123)(88,122,98,132)(89,131,99,121)(90,140,100,130)(101,157,111,147)(102,146,112,156)(103,155,113,145)(104,144,114,154)(105,153,115,143)(106,142,116,152)(107,151,117,141)(108,160,118,150)(109,149,119,159)(110,158,120,148) );
G=PermutationGroup([[(1,76,11,66),(2,147,12,157),(3,78,13,68),(4,149,14,159),(5,80,15,70),(6,151,16,141),(7,62,17,72),(8,153,18,143),(9,64,19,74),(10,155,20,145),(21,98,31,88),(22,45,32,55),(23,100,33,90),(24,47,34,57),(25,82,35,92),(26,49,36,59),(27,84,37,94),(28,51,38,41),(29,86,39,96),(30,53,40,43),(42,135,52,125),(44,137,54,127),(46,139,56,129),(48,121,58,131),(50,123,60,133),(61,108,71,118),(63,110,73,120),(65,112,75,102),(67,114,77,104),(69,116,79,106),(81,140,91,130),(83,122,93,132),(85,124,95,134),(87,126,97,136),(89,128,99,138),(101,154,111,144),(103,156,113,146),(105,158,115,148),(107,160,117,150),(109,142,119,152)], [(1,52,113,86),(2,43,114,97),(3,54,115,88),(4,45,116,99),(5,56,117,90),(6,47,118,81),(7,58,119,92),(8,49,120,83),(9,60,101,94),(10,51,102,85),(11,42,103,96),(12,53,104,87),(13,44,105,98),(14,55,106,89),(15,46,107,100),(16,57,108,91),(17,48,109,82),(18,59,110,93),(19,50,111,84),(20,41,112,95),(21,78,127,148),(22,69,128,159),(23,80,129,150),(24,71,130,141),(25,62,131,152),(26,73,132,143),(27,64,133,154),(28,75,134,145),(29,66,135,156),(30,77,136,147),(31,68,137,158),(32,79,138,149),(33,70,139,160),(34,61,140,151),(35,72,121,142),(36,63,122,153),(37,74,123,144),(38,65,124,155),(39,76,125,146),(40,67,126,157)], [(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)], [(1,75,11,65),(2,64,12,74),(3,73,13,63),(4,62,14,72),(5,71,15,61),(6,80,16,70),(7,69,17,79),(8,78,18,68),(9,67,19,77),(10,76,20,66),(21,49,31,59),(22,58,32,48),(23,47,33,57),(24,56,34,46),(25,45,35,55),(26,54,36,44),(27,43,37,53),(28,52,38,42),(29,41,39,51),(30,50,40,60),(81,139,91,129),(82,128,92,138),(83,137,93,127),(84,126,94,136),(85,135,95,125),(86,124,96,134),(87,133,97,123),(88,122,98,132),(89,131,99,121),(90,140,100,130),(101,157,111,147),(102,146,112,156),(103,155,113,145),(104,144,114,154),(105,153,115,143),(106,142,116,152),(107,151,117,141),(108,160,118,150),(109,149,119,159),(110,158,120,148)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | ··· | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ 1+4 | Q8⋊2D5 | D4⋊8D10 |
| kernel | C42.156D10 | C42⋊D5 | C20⋊4D4 | D20⋊8C4 | D10.13D4 | C4⋊D20 | C5×C42.C2 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 6 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 8 |
Matrix representation of C42.156D10 ►in GL8(𝔽41)
| 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 | 11 | 13 | 21 | 21 |
| 0 | 0 | 0 | 0 | 19 | 30 | 40 | 38 |
| 0 | 0 | 0 | 0 | 6 | 1 | 2 | 28 |
| 0 | 0 | 0 | 0 | 39 | 40 | 13 | 39 |
| 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 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 2 |
| 40 | 1 | 8 | 8 | 0 | 0 | 0 | 0 |
| 40 | 33 | 33 | 34 | 0 | 0 | 0 | 0 |
| 8 | 8 | 1 | 40 | 0 | 0 | 0 | 0 |
| 33 | 34 | 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 | 18 | 20 |
| 0 | 0 | 0 | 0 | 22 | 26 | 21 | 21 |
| 33 | 33 | 40 | 1 | 0 | 0 | 0 | 0 |
| 7 | 8 | 8 | 1 | 0 | 0 | 0 | 0 |
| 1 | 40 | 33 | 33 | 0 | 0 | 0 | 0 |
| 33 | 40 | 7 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 9 | 1 | 6 |
| 0 | 0 | 0 | 0 | 19 | 19 | 40 | 34 |
| 0 | 0 | 0 | 0 | 14 | 12 | 32 | 0 |
| 0 | 0 | 0 | 0 | 39 | 39 | 13 | 9 |
G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,19,6,39,0,0,0,0,13,30,1,40,0,0,0,0,21,40,2,13,0,0,0,0,21,38,28,39],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,13,2],[40,40,8,33,0,0,0,0,1,33,8,34,0,0,0,0,8,33,1,1,0,0,0,0,8,34,40,8,0,0,0,0,0,0,0,0,3,37,13,22,0,0,0,0,21,40,4,26,0,0,0,0,0,0,18,21,0,0,0,0,0,0,20,21],[33,7,1,33,0,0,0,0,33,8,40,40,0,0,0,0,40,8,33,7,0,0,0,0,1,1,33,8,0,0,0,0,0,0,0,0,22,19,14,39,0,0,0,0,9,19,12,39,0,0,0,0,1,40,32,13,0,0,0,0,6,34,0,9] >;
C42.156D10 in GAP, Magma, Sage, TeX
C_4^2._{156}D_{10} % in TeX
G:=Group("C4^2.156D10"); // GroupNames label
G:=SmallGroup(320,1370);
// by ID
G=gap.SmallGroup(320,1370);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations