metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.56D10, Q8⋊D5⋊9C4, (C4×Q8)⋊2D5, Q8⋊4(C4×D5), (Q8×C20)⋊2C2, C4⋊C4.252D10, (C4×D20).15C2, D20.31(C2×C4), (C2×C20).258D4, C10.104(C4×D4), C20.59(C4○D4), C4.41(C4○D20), Q8⋊Dic5⋊11C2, C10.D8⋊33C2, C5⋊7(SD16⋊C4), (C4×C20).97C22, C20.61(C22×C4), (C2×Q8).159D10, C42.D5⋊7C2, D20⋊6C4.10C2, C2.4(D4⋊D10), (C2×C20).346C23, C10.111(C8⋊C22), C2.3(C20.C23), (C2×D20).247C22, C10.87(C8.C22), C4⋊Dic5.331C22, (Q8×C10).194C22, C4.26(C2×C4×D5), C5⋊2C8⋊10(C2×C4), (C5×Q8)⋊18(C2×C4), C2.20(C4×C5⋊D4), (C2×Q8⋊D5).4C2, (C2×C10).477(C2×D4), C22.80(C2×C5⋊D4), (C2×C4).221(C5⋊D4), (C5×C4⋊C4).283C22, (C2×C4).446(C22×D5), (C2×C5⋊2C8).100C22, SmallGroup(320,653)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.56D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >
Subgroups: 454 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, SD16⋊C4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, Q8⋊D5, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C42.D5, C10.D8, D20⋊6C4, Q8⋊Dic5, C4×D20, C2×Q8⋊D5, Q8×C20, C42.56D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C8.C22, C4×D5, C5⋊D4, C22×D5, SD16⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C20.C23, D4⋊D10, C42.56D10
►On 160 points
Generators in S160
(1 99 119 153)(2 90 120 144)(3 81 101 155)(4 92 102 146)(5 83 103 157)(6 94 104 148)(7 85 105 159)(8 96 106 150)(9 87 107 141)(10 98 108 152)(11 89 109 143)(12 100 110 154)(13 91 111 145)(14 82 112 156)(15 93 113 147)(16 84 114 158)(17 95 115 149)(18 86 116 160)(19 97 117 151)(20 88 118 142)(21 62 133 56)(22 73 134 47)(23 64 135 58)(24 75 136 49)(25 66 137 60)(26 77 138 51)(27 68 139 42)(28 79 140 53)(29 70 121 44)(30 61 122 55)(31 72 123 46)(32 63 124 57)(33 74 125 48)(34 65 126 59)(35 76 127 50)(36 67 128 41)(37 78 129 52)(38 69 130 43)(39 80 131 54)(40 71 132 45)
(1 37 11 27)(2 28 12 38)(3 39 13 29)(4 30 14 40)(5 21 15 31)(6 32 16 22)(7 23 17 33)(8 34 18 24)(9 25 19 35)(10 36 20 26)(41 142 51 152)(42 153 52 143)(43 144 53 154)(44 155 54 145)(45 146 55 156)(46 157 56 147)(47 148 57 158)(48 159 58 149)(49 150 59 160)(50 141 60 151)(61 82 71 92)(62 93 72 83)(63 84 73 94)(64 95 74 85)(65 86 75 96)(66 97 76 87)(67 88 77 98)(68 99 78 89)(69 90 79 100)(70 81 80 91)(101 131 111 121)(102 122 112 132)(103 133 113 123)(104 124 114 134)(105 135 115 125)(106 126 116 136)(107 137 117 127)(108 128 118 138)(109 139 119 129)(110 130 120 140)
(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 10 37 36 11 20 27 26)(2 25 28 19 12 35 38 9)(3 8 39 34 13 18 29 24)(4 23 30 17 14 33 40 7)(5 6 21 32 15 16 31 22)(41 153 142 52 51 143 152 42)(43 151 144 50 53 141 154 60)(44 59 155 160 54 49 145 150)(45 149 146 48 55 159 156 58)(46 57 157 158 56 47 147 148)(61 85 82 64 71 95 92 74)(62 73 93 94 72 63 83 84)(65 81 86 80 75 91 96 70)(66 69 97 90 76 79 87 100)(67 99 88 78 77 89 98 68)(101 106 131 126 111 116 121 136)(102 135 122 115 112 125 132 105)(103 104 133 124 113 114 123 134)(107 120 137 140 117 110 127 130)(108 129 128 109 118 139 138 119)
G:=sub<Sym(160)| (1,99,119,153)(2,90,120,144)(3,81,101,155)(4,92,102,146)(5,83,103,157)(6,94,104,148)(7,85,105,159)(8,96,106,150)(9,87,107,141)(10,98,108,152)(11,89,109,143)(12,100,110,154)(13,91,111,145)(14,82,112,156)(15,93,113,147)(16,84,114,158)(17,95,115,149)(18,86,116,160)(19,97,117,151)(20,88,118,142)(21,62,133,56)(22,73,134,47)(23,64,135,58)(24,75,136,49)(25,66,137,60)(26,77,138,51)(27,68,139,42)(28,79,140,53)(29,70,121,44)(30,61,122,55)(31,72,123,46)(32,63,124,57)(33,74,125,48)(34,65,126,59)(35,76,127,50)(36,67,128,41)(37,78,129,52)(38,69,130,43)(39,80,131,54)(40,71,132,45), (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,142,51,152)(42,153,52,143)(43,144,53,154)(44,155,54,145)(45,146,55,156)(46,157,56,147)(47,148,57,158)(48,159,58,149)(49,150,59,160)(50,141,60,151)(61,82,71,92)(62,93,72,83)(63,84,73,94)(64,95,74,85)(65,86,75,96)(66,97,76,87)(67,88,77,98)(68,99,78,89)(69,90,79,100)(70,81,80,91)(101,131,111,121)(102,122,112,132)(103,133,113,123)(104,124,114,134)(105,135,115,125)(106,126,116,136)(107,137,117,127)(108,128,118,138)(109,139,119,129)(110,130,120,140), (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), (1,10,37,36,11,20,27,26)(2,25,28,19,12,35,38,9)(3,8,39,34,13,18,29,24)(4,23,30,17,14,33,40,7)(5,6,21,32,15,16,31,22)(41,153,142,52,51,143,152,42)(43,151,144,50,53,141,154,60)(44,59,155,160,54,49,145,150)(45,149,146,48,55,159,156,58)(46,57,157,158,56,47,147,148)(61,85,82,64,71,95,92,74)(62,73,93,94,72,63,83,84)(65,81,86,80,75,91,96,70)(66,69,97,90,76,79,87,100)(67,99,88,78,77,89,98,68)(101,106,131,126,111,116,121,136)(102,135,122,115,112,125,132,105)(103,104,133,124,113,114,123,134)(107,120,137,140,117,110,127,130)(108,129,128,109,118,139,138,119)>;
G:=Group( (1,99,119,153)(2,90,120,144)(3,81,101,155)(4,92,102,146)(5,83,103,157)(6,94,104,148)(7,85,105,159)(8,96,106,150)(9,87,107,141)(10,98,108,152)(11,89,109,143)(12,100,110,154)(13,91,111,145)(14,82,112,156)(15,93,113,147)(16,84,114,158)(17,95,115,149)(18,86,116,160)(19,97,117,151)(20,88,118,142)(21,62,133,56)(22,73,134,47)(23,64,135,58)(24,75,136,49)(25,66,137,60)(26,77,138,51)(27,68,139,42)(28,79,140,53)(29,70,121,44)(30,61,122,55)(31,72,123,46)(32,63,124,57)(33,74,125,48)(34,65,126,59)(35,76,127,50)(36,67,128,41)(37,78,129,52)(38,69,130,43)(39,80,131,54)(40,71,132,45), (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,142,51,152)(42,153,52,143)(43,144,53,154)(44,155,54,145)(45,146,55,156)(46,157,56,147)(47,148,57,158)(48,159,58,149)(49,150,59,160)(50,141,60,151)(61,82,71,92)(62,93,72,83)(63,84,73,94)(64,95,74,85)(65,86,75,96)(66,97,76,87)(67,88,77,98)(68,99,78,89)(69,90,79,100)(70,81,80,91)(101,131,111,121)(102,122,112,132)(103,133,113,123)(104,124,114,134)(105,135,115,125)(106,126,116,136)(107,137,117,127)(108,128,118,138)(109,139,119,129)(110,130,120,140), (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), (1,10,37,36,11,20,27,26)(2,25,28,19,12,35,38,9)(3,8,39,34,13,18,29,24)(4,23,30,17,14,33,40,7)(5,6,21,32,15,16,31,22)(41,153,142,52,51,143,152,42)(43,151,144,50,53,141,154,60)(44,59,155,160,54,49,145,150)(45,149,146,48,55,159,156,58)(46,57,157,158,56,47,147,148)(61,85,82,64,71,95,92,74)(62,73,93,94,72,63,83,84)(65,81,86,80,75,91,96,70)(66,69,97,90,76,79,87,100)(67,99,88,78,77,89,98,68)(101,106,131,126,111,116,121,136)(102,135,122,115,112,125,132,105)(103,104,133,124,113,114,123,134)(107,120,137,140,117,110,127,130)(108,129,128,109,118,139,138,119) );
G=PermutationGroup([[(1,99,119,153),(2,90,120,144),(3,81,101,155),(4,92,102,146),(5,83,103,157),(6,94,104,148),(7,85,105,159),(8,96,106,150),(9,87,107,141),(10,98,108,152),(11,89,109,143),(12,100,110,154),(13,91,111,145),(14,82,112,156),(15,93,113,147),(16,84,114,158),(17,95,115,149),(18,86,116,160),(19,97,117,151),(20,88,118,142),(21,62,133,56),(22,73,134,47),(23,64,135,58),(24,75,136,49),(25,66,137,60),(26,77,138,51),(27,68,139,42),(28,79,140,53),(29,70,121,44),(30,61,122,55),(31,72,123,46),(32,63,124,57),(33,74,125,48),(34,65,126,59),(35,76,127,50),(36,67,128,41),(37,78,129,52),(38,69,130,43),(39,80,131,54),(40,71,132,45)], [(1,37,11,27),(2,28,12,38),(3,39,13,29),(4,30,14,40),(5,21,15,31),(6,32,16,22),(7,23,17,33),(8,34,18,24),(9,25,19,35),(10,36,20,26),(41,142,51,152),(42,153,52,143),(43,144,53,154),(44,155,54,145),(45,146,55,156),(46,157,56,147),(47,148,57,158),(48,159,58,149),(49,150,59,160),(50,141,60,151),(61,82,71,92),(62,93,72,83),(63,84,73,94),(64,95,74,85),(65,86,75,96),(66,97,76,87),(67,88,77,98),(68,99,78,89),(69,90,79,100),(70,81,80,91),(101,131,111,121),(102,122,112,132),(103,133,113,123),(104,124,114,134),(105,135,115,125),(106,126,116,136),(107,137,117,127),(108,128,118,138),(109,139,119,129),(110,130,120,140)], [(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)], [(1,10,37,36,11,20,27,26),(2,25,28,19,12,35,38,9),(3,8,39,34,13,18,29,24),(4,23,30,17,14,33,40,7),(5,6,21,32,15,16,31,22),(41,153,142,52,51,143,152,42),(43,151,144,50,53,141,154,60),(44,59,155,160,54,49,145,150),(45,149,146,48,55,159,156,58),(46,57,157,158,56,47,147,148),(61,85,82,64,71,95,92,74),(62,73,93,94,72,63,83,84),(65,81,86,80,75,91,96,70),(66,69,97,90,76,79,87,100),(67,99,88,78,77,89,98,68),(101,106,131,126,111,116,121,136),(102,135,122,115,112,125,132,105),(103,104,133,124,113,114,123,134),(107,120,137,140,117,110,127,130),(108,129,128,109,118,139,138,119)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C4×D5 | C4○D20 | C8⋊C22 | C8.C22 | C20.C23 | D4⋊D10 |
| kernel | C42.56D10 | C42.D5 | C10.D8 | D20⋊6C4 | Q8⋊Dic5 | C4×D20 | C2×Q8⋊D5 | Q8×C20 | Q8⋊D5 | C2×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C2×C4 | Q8 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C42.56D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 21 | 0 |
| 0 | 0 | 6 | 1 | 28 | 39 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 6 | 0 | 22 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 |
| 0 | 0 | 26 | 0 | 13 | 5 |
| 0 | 0 | 1 | 27 | 7 | 28 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 27 | 35 | 33 |
| 0 | 0 | 35 | 35 | 11 | 20 |
| 0 | 0 | 21 | 15 | 40 | 26 |
| 0 | 0 | 10 | 39 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 14 | 11 | 8 |
| 0 | 0 | 6 | 6 | 30 | 21 |
| 0 | 0 | 37 | 26 | 36 | 15 |
| 0 | 0 | 0 | 2 | 13 | 23 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,6,0,6,0,0,0,1,0,0,0,0,21,28,40,22,0,0,0,39,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,26,1,0,0,9,40,0,27,0,0,0,0,13,7,0,0,0,0,5,28],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,30,35,21,10,0,0,27,35,15,39,0,0,35,11,40,0,0,0,33,20,26,18],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,17,6,37,0,0,0,14,6,26,2,0,0,11,30,36,13,0,0,8,21,15,23] >;
C42.56D10 in GAP, Magma, Sage, TeX
C_4^2._{56}D_{10} % in TeX
G:=Group("C4^2.56D10"); // GroupNames label
G:=SmallGroup(320,653);
// by ID
G=gap.SmallGroup(320,653);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,387,58,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations