metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.61D10, Dic10.23D4, C4.49(D4×D5), C20⋊3C8⋊28C2, C20.23(C2×D4), (C2×D4).44D10, (C2×C20).269D4, (C2×Q8).34D10, C5⋊5(Q8.D4), C4.4D4.4D5, (C4×Dic10)⋊21C2, C20.65(C4○D4), C4.1(D4⋊2D5), Q8⋊Dic5⋊19C2, C10.103(C4○D8), C2.10(C20⋊2D4), (C2×C20).372C23, (C4×C20).103C22, D4⋊Dic5.11C2, (D4×C10).60C22, (Q8×C10).52C22, C10.101(C4⋊D4), C4⋊Dic5.340C22, C2.22(D4.8D10), C2.17(D4.9D10), C10.118(C8.C22), (C2×Dic10).280C22, (C2×C5⋊Q16)⋊12C2, (C2×D4.D5).6C2, (C2×C10).503(C2×D4), (C2×C4).59(C5⋊D4), (C5×C4.4D4).2C2, (C2×C4).472(C22×D5), C22.178(C2×C5⋊D4), (C2×C5⋊2C8).119C22, SmallGroup(320,681)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.61D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 374 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, Q8.D4, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C5⋊Q16, C4×C20, C5×C22⋊C4, C2×Dic10, D4×C10, Q8×C10, C20⋊3C8, D4⋊Dic5, Q8⋊Dic5, C4×Dic10, C2×D4.D5, C2×C5⋊Q16, C5×C4.4D4, C42.61D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, C5⋊D4, C22×D5, Q8.D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4, D4.8D10, D4.9D10, C42.61D10
►On 160 points
Generators in S160
(1 70 18 59)(2 66 19 55)(3 62 20 51)(4 68 16 57)(5 64 17 53)(6 52 15 63)(7 58 11 69)(8 54 12 65)(9 60 13 61)(10 56 14 67)(21 81 38 86)(22 79 39 74)(23 83 40 88)(24 71 31 76)(25 85 32 90)(26 73 33 78)(27 87 34 82)(28 75 35 80)(29 89 36 84)(30 77 37 72)(41 117 125 95)(42 131 126 103)(43 119 127 97)(44 133 128 105)(45 111 129 99)(46 135 130 107)(47 113 121 91)(48 137 122 109)(49 115 123 93)(50 139 124 101)(92 154 114 149)(94 156 116 141)(96 158 118 143)(98 160 120 145)(100 152 112 147)(102 157 140 142)(104 159 132 144)(106 151 134 146)(108 153 136 148)(110 155 138 150)
(1 26 8 21)(2 22 9 27)(3 28 10 23)(4 24 6 29)(5 30 7 25)(11 32 17 37)(12 38 18 33)(13 34 19 39)(14 40 20 35)(15 36 16 31)(41 130 142 152)(42 153 143 121)(43 122 144 154)(44 155 145 123)(45 124 146 156)(46 157 147 125)(47 126 148 158)(48 159 149 127)(49 128 150 160)(50 151 141 129)(51 80 67 88)(52 89 68 71)(53 72 69 90)(54 81 70 73)(55 74 61 82)(56 83 62 75)(57 76 63 84)(58 85 64 77)(59 78 65 86)(60 87 66 79)(91 131 136 96)(92 97 137 132)(93 133 138 98)(94 99 139 134)(95 135 140 100)(101 106 116 111)(102 112 117 107)(103 108 118 113)(104 114 119 109)(105 110 120 115)
(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)
(1 139 33 106 8 94 38 111)(2 133 39 110 9 98 34 115)(3 137 35 104 10 92 40 119)(4 131 31 108 6 96 36 113)(5 135 37 102 7 100 32 117)(11 112 25 95 17 107 30 140)(12 116 21 99 18 101 26 134)(13 120 27 93 19 105 22 138)(14 114 23 97 20 109 28 132)(15 118 29 91 16 103 24 136)(41 58 46 90 142 64 147 72)(42 84 148 68 143 76 47 52)(43 56 48 88 144 62 149 80)(44 82 150 66 145 74 49 60)(45 54 50 86 146 70 141 78)(51 154 75 127 67 122 83 159)(53 152 77 125 69 130 85 157)(55 160 79 123 61 128 87 155)(57 158 71 121 63 126 89 153)(59 156 73 129 65 124 81 151)
G:=sub<Sym(160)| (1,70,18,59)(2,66,19,55)(3,62,20,51)(4,68,16,57)(5,64,17,53)(6,52,15,63)(7,58,11,69)(8,54,12,65)(9,60,13,61)(10,56,14,67)(21,81,38,86)(22,79,39,74)(23,83,40,88)(24,71,31,76)(25,85,32,90)(26,73,33,78)(27,87,34,82)(28,75,35,80)(29,89,36,84)(30,77,37,72)(41,117,125,95)(42,131,126,103)(43,119,127,97)(44,133,128,105)(45,111,129,99)(46,135,130,107)(47,113,121,91)(48,137,122,109)(49,115,123,93)(50,139,124,101)(92,154,114,149)(94,156,116,141)(96,158,118,143)(98,160,120,145)(100,152,112,147)(102,157,140,142)(104,159,132,144)(106,151,134,146)(108,153,136,148)(110,155,138,150), (1,26,8,21)(2,22,9,27)(3,28,10,23)(4,24,6,29)(5,30,7,25)(11,32,17,37)(12,38,18,33)(13,34,19,39)(14,40,20,35)(15,36,16,31)(41,130,142,152)(42,153,143,121)(43,122,144,154)(44,155,145,123)(45,124,146,156)(46,157,147,125)(47,126,148,158)(48,159,149,127)(49,128,150,160)(50,151,141,129)(51,80,67,88)(52,89,68,71)(53,72,69,90)(54,81,70,73)(55,74,61,82)(56,83,62,75)(57,76,63,84)(58,85,64,77)(59,78,65,86)(60,87,66,79)(91,131,136,96)(92,97,137,132)(93,133,138,98)(94,99,139,134)(95,135,140,100)(101,106,116,111)(102,112,117,107)(103,108,118,113)(104,114,119,109)(105,110,120,115), (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), (1,139,33,106,8,94,38,111)(2,133,39,110,9,98,34,115)(3,137,35,104,10,92,40,119)(4,131,31,108,6,96,36,113)(5,135,37,102,7,100,32,117)(11,112,25,95,17,107,30,140)(12,116,21,99,18,101,26,134)(13,120,27,93,19,105,22,138)(14,114,23,97,20,109,28,132)(15,118,29,91,16,103,24,136)(41,58,46,90,142,64,147,72)(42,84,148,68,143,76,47,52)(43,56,48,88,144,62,149,80)(44,82,150,66,145,74,49,60)(45,54,50,86,146,70,141,78)(51,154,75,127,67,122,83,159)(53,152,77,125,69,130,85,157)(55,160,79,123,61,128,87,155)(57,158,71,121,63,126,89,153)(59,156,73,129,65,124,81,151)>;
G:=Group( (1,70,18,59)(2,66,19,55)(3,62,20,51)(4,68,16,57)(5,64,17,53)(6,52,15,63)(7,58,11,69)(8,54,12,65)(9,60,13,61)(10,56,14,67)(21,81,38,86)(22,79,39,74)(23,83,40,88)(24,71,31,76)(25,85,32,90)(26,73,33,78)(27,87,34,82)(28,75,35,80)(29,89,36,84)(30,77,37,72)(41,117,125,95)(42,131,126,103)(43,119,127,97)(44,133,128,105)(45,111,129,99)(46,135,130,107)(47,113,121,91)(48,137,122,109)(49,115,123,93)(50,139,124,101)(92,154,114,149)(94,156,116,141)(96,158,118,143)(98,160,120,145)(100,152,112,147)(102,157,140,142)(104,159,132,144)(106,151,134,146)(108,153,136,148)(110,155,138,150), (1,26,8,21)(2,22,9,27)(3,28,10,23)(4,24,6,29)(5,30,7,25)(11,32,17,37)(12,38,18,33)(13,34,19,39)(14,40,20,35)(15,36,16,31)(41,130,142,152)(42,153,143,121)(43,122,144,154)(44,155,145,123)(45,124,146,156)(46,157,147,125)(47,126,148,158)(48,159,149,127)(49,128,150,160)(50,151,141,129)(51,80,67,88)(52,89,68,71)(53,72,69,90)(54,81,70,73)(55,74,61,82)(56,83,62,75)(57,76,63,84)(58,85,64,77)(59,78,65,86)(60,87,66,79)(91,131,136,96)(92,97,137,132)(93,133,138,98)(94,99,139,134)(95,135,140,100)(101,106,116,111)(102,112,117,107)(103,108,118,113)(104,114,119,109)(105,110,120,115), (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), (1,139,33,106,8,94,38,111)(2,133,39,110,9,98,34,115)(3,137,35,104,10,92,40,119)(4,131,31,108,6,96,36,113)(5,135,37,102,7,100,32,117)(11,112,25,95,17,107,30,140)(12,116,21,99,18,101,26,134)(13,120,27,93,19,105,22,138)(14,114,23,97,20,109,28,132)(15,118,29,91,16,103,24,136)(41,58,46,90,142,64,147,72)(42,84,148,68,143,76,47,52)(43,56,48,88,144,62,149,80)(44,82,150,66,145,74,49,60)(45,54,50,86,146,70,141,78)(51,154,75,127,67,122,83,159)(53,152,77,125,69,130,85,157)(55,160,79,123,61,128,87,155)(57,158,71,121,63,126,89,153)(59,156,73,129,65,124,81,151) );
G=PermutationGroup([[(1,70,18,59),(2,66,19,55),(3,62,20,51),(4,68,16,57),(5,64,17,53),(6,52,15,63),(7,58,11,69),(8,54,12,65),(9,60,13,61),(10,56,14,67),(21,81,38,86),(22,79,39,74),(23,83,40,88),(24,71,31,76),(25,85,32,90),(26,73,33,78),(27,87,34,82),(28,75,35,80),(29,89,36,84),(30,77,37,72),(41,117,125,95),(42,131,126,103),(43,119,127,97),(44,133,128,105),(45,111,129,99),(46,135,130,107),(47,113,121,91),(48,137,122,109),(49,115,123,93),(50,139,124,101),(92,154,114,149),(94,156,116,141),(96,158,118,143),(98,160,120,145),(100,152,112,147),(102,157,140,142),(104,159,132,144),(106,151,134,146),(108,153,136,148),(110,155,138,150)], [(1,26,8,21),(2,22,9,27),(3,28,10,23),(4,24,6,29),(5,30,7,25),(11,32,17,37),(12,38,18,33),(13,34,19,39),(14,40,20,35),(15,36,16,31),(41,130,142,152),(42,153,143,121),(43,122,144,154),(44,155,145,123),(45,124,146,156),(46,157,147,125),(47,126,148,158),(48,159,149,127),(49,128,150,160),(50,151,141,129),(51,80,67,88),(52,89,68,71),(53,72,69,90),(54,81,70,73),(55,74,61,82),(56,83,62,75),(57,76,63,84),(58,85,64,77),(59,78,65,86),(60,87,66,79),(91,131,136,96),(92,97,137,132),(93,133,138,98),(94,99,139,134),(95,135,140,100),(101,106,116,111),(102,112,117,107),(103,108,118,113),(104,114,119,109),(105,110,120,115)], [(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)], [(1,139,33,106,8,94,38,111),(2,133,39,110,9,98,34,115),(3,137,35,104,10,92,40,119),(4,131,31,108,6,96,36,113),(5,135,37,102,7,100,32,117),(11,112,25,95,17,107,30,140),(12,116,21,99,18,101,26,134),(13,120,27,93,19,105,22,138),(14,114,23,97,20,109,28,132),(15,118,29,91,16,103,24,136),(41,58,46,90,142,64,147,72),(42,84,148,68,143,76,47,52),(43,56,48,88,144,62,149,80),(44,82,150,66,145,74,49,60),(45,54,50,86,146,70,141,78),(51,154,75,127,67,122,83,159),(53,152,77,125,69,130,85,157),(55,160,79,123,61,128,87,155),(57,158,71,121,63,126,89,153),(59,156,73,129,65,124,81,151)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 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 | 2 | 2 | 4 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C8.C22 | D4×D5 | D4⋊2D5 | D4.8D10 | D4.9D10 |
| kernel | C42.61D10 | C20⋊3C8 | D4⋊Dic5 | Q8⋊Dic5 | C4×Dic10 | C2×D4.D5 | C2×C5⋊Q16 | C5×C4.4D4 | Dic10 | C2×C20 | C4.4D4 | C20 | C42 | C2×D4 | C2×Q8 | C10 | C2×C4 | C10 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C42.61D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 9 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 4 | 40 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 13 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 10 | 18 | 0 | 0 | 0 | 0 |
| 24 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 40 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 34 | 24 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,9,0,0,0,0,18,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,20,40],[16,13,0,0,0,0,0,23,0,0,0,0,0,0,1,9,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,20,40],[10,24,0,0,0,0,18,31,0,0,0,0,0,0,9,40,0,0,0,0,0,32,0,0,0,0,0,0,0,34,0,0,0,0,6,24] >;
C42.61D10 in GAP, Magma, Sage, TeX
C_4^2._{61}D_{10} % in TeX
G:=Group("C4^2.61D10"); // GroupNames label
G:=SmallGroup(320,681);
// by ID
G=gap.SmallGroup(320,681);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations