metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.65D10, C4.13(D4×D5), C5⋊2C8.7D4, C20.27(C2×D4), (C2×C20).83D4, C5⋊2(C8.2D4), C20⋊2Q8⋊18C2, (C2×D4).50D10, (C2×Q8).40D10, C4.4D4.7D5, C2.10(C20⋊D4), C10.19(C4⋊1D4), (C4×C20).109C22, (C2×C20).378C23, (D4×C10).66C22, C42.D5⋊10C2, (Q8×C10).58C22, C2.19(D4.9D10), C10.120(C8.C22), (C2×Dic10).112C22, (C2×C5⋊Q16)⋊14C2, (C2×D4.D5).7C2, (C2×C10).509(C2×D4), (C2×C4).63(C5⋊D4), (C5×C4.4D4).5C2, (C2×C4).478(C22×D5), C22.184(C2×C5⋊D4), (C2×C5⋊2C8).123C22, SmallGroup(320,687)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.65D10
G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 446 in 124 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C8.2D4, C2×C5⋊2C8, C4⋊Dic5, D4.D5, C5⋊Q16, C4×C20, C5×C22⋊C4, C2×Dic10, D4×C10, Q8×C10, C42.D5, C20⋊2Q8, C2×D4.D5, C2×C5⋊Q16, C5×C4.4D4, C42.65D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8.C22, C5⋊D4, C22×D5, C8.2D4, D4×D5, C2×C5⋊D4, C20⋊D4, D4.9D10, C42.65D10
►On 160 points
Generators in S160
(1 43 11 51)(2 49 12 57)(3 45 13 53)(4 41 14 59)(5 47 15 55)(6 56 16 48)(7 52 17 44)(8 58 18 50)(9 54 19 46)(10 60 20 42)(21 160 112 91)(22 150 113 128)(23 152 114 93)(24 142 115 130)(25 154 116 95)(26 144 117 122)(27 156 118 97)(28 146 119 124)(29 158 120 99)(30 148 111 126)(31 69 136 64)(32 77 137 72)(33 61 138 66)(34 79 139 74)(35 63 140 68)(36 71 131 76)(37 65 132 70)(38 73 133 78)(39 67 134 62)(40 75 135 80)(81 94 108 153)(82 121 109 143)(83 96 110 155)(84 123 101 145)(85 98 102 157)(86 125 103 147)(87 100 104 159)(88 127 105 149)(89 92 106 151)(90 129 107 141)
(1 131 6 136)(2 137 7 132)(3 133 8 138)(4 139 9 134)(5 135 10 140)(11 36 16 31)(12 32 17 37)(13 38 18 33)(14 34 19 39)(15 40 20 35)(21 26 88 83)(22 84 89 27)(23 28 90 85)(24 86 81 29)(25 30 82 87)(41 74 54 62)(42 63 55 75)(43 76 56 64)(44 65 57 77)(45 78 58 66)(46 67 59 79)(47 80 60 68)(48 69 51 71)(49 72 52 70)(50 61 53 73)(91 122 149 155)(92 156 150 123)(93 124 141 157)(94 158 142 125)(95 126 143 159)(96 160 144 127)(97 128 145 151)(98 152 146 129)(99 130 147 153)(100 154 148 121)(101 106 118 113)(102 114 119 107)(103 108 120 115)(104 116 111 109)(105 110 112 117)
(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 86 131 81 6 29 136 24)(2 90 137 85 7 23 132 28)(3 84 133 89 8 27 138 22)(4 88 139 83 9 21 134 26)(5 82 135 87 10 25 140 30)(11 103 36 108 16 120 31 115)(12 107 32 102 17 114 37 119)(13 101 38 106 18 118 33 113)(14 105 34 110 19 112 39 117)(15 109 40 104 20 116 35 111)(41 160 74 144 54 127 62 96)(42 143 63 159 55 95 75 126)(43 158 76 142 56 125 64 94)(44 141 65 157 57 93 77 124)(45 156 78 150 58 123 66 92)(46 149 67 155 59 91 79 122)(47 154 80 148 60 121 68 100)(48 147 69 153 51 99 71 130)(49 152 72 146 52 129 70 98)(50 145 61 151 53 97 73 128)
G:=sub<Sym(160)| (1,43,11,51)(2,49,12,57)(3,45,13,53)(4,41,14,59)(5,47,15,55)(6,56,16,48)(7,52,17,44)(8,58,18,50)(9,54,19,46)(10,60,20,42)(21,160,112,91)(22,150,113,128)(23,152,114,93)(24,142,115,130)(25,154,116,95)(26,144,117,122)(27,156,118,97)(28,146,119,124)(29,158,120,99)(30,148,111,126)(31,69,136,64)(32,77,137,72)(33,61,138,66)(34,79,139,74)(35,63,140,68)(36,71,131,76)(37,65,132,70)(38,73,133,78)(39,67,134,62)(40,75,135,80)(81,94,108,153)(82,121,109,143)(83,96,110,155)(84,123,101,145)(85,98,102,157)(86,125,103,147)(87,100,104,159)(88,127,105,149)(89,92,106,151)(90,129,107,141), (1,131,6,136)(2,137,7,132)(3,133,8,138)(4,139,9,134)(5,135,10,140)(11,36,16,31)(12,32,17,37)(13,38,18,33)(14,34,19,39)(15,40,20,35)(21,26,88,83)(22,84,89,27)(23,28,90,85)(24,86,81,29)(25,30,82,87)(41,74,54,62)(42,63,55,75)(43,76,56,64)(44,65,57,77)(45,78,58,66)(46,67,59,79)(47,80,60,68)(48,69,51,71)(49,72,52,70)(50,61,53,73)(91,122,149,155)(92,156,150,123)(93,124,141,157)(94,158,142,125)(95,126,143,159)(96,160,144,127)(97,128,145,151)(98,152,146,129)(99,130,147,153)(100,154,148,121)(101,106,118,113)(102,114,119,107)(103,108,120,115)(104,116,111,109)(105,110,112,117), (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), (1,86,131,81,6,29,136,24)(2,90,137,85,7,23,132,28)(3,84,133,89,8,27,138,22)(4,88,139,83,9,21,134,26)(5,82,135,87,10,25,140,30)(11,103,36,108,16,120,31,115)(12,107,32,102,17,114,37,119)(13,101,38,106,18,118,33,113)(14,105,34,110,19,112,39,117)(15,109,40,104,20,116,35,111)(41,160,74,144,54,127,62,96)(42,143,63,159,55,95,75,126)(43,158,76,142,56,125,64,94)(44,141,65,157,57,93,77,124)(45,156,78,150,58,123,66,92)(46,149,67,155,59,91,79,122)(47,154,80,148,60,121,68,100)(48,147,69,153,51,99,71,130)(49,152,72,146,52,129,70,98)(50,145,61,151,53,97,73,128)>;
G:=Group( (1,43,11,51)(2,49,12,57)(3,45,13,53)(4,41,14,59)(5,47,15,55)(6,56,16,48)(7,52,17,44)(8,58,18,50)(9,54,19,46)(10,60,20,42)(21,160,112,91)(22,150,113,128)(23,152,114,93)(24,142,115,130)(25,154,116,95)(26,144,117,122)(27,156,118,97)(28,146,119,124)(29,158,120,99)(30,148,111,126)(31,69,136,64)(32,77,137,72)(33,61,138,66)(34,79,139,74)(35,63,140,68)(36,71,131,76)(37,65,132,70)(38,73,133,78)(39,67,134,62)(40,75,135,80)(81,94,108,153)(82,121,109,143)(83,96,110,155)(84,123,101,145)(85,98,102,157)(86,125,103,147)(87,100,104,159)(88,127,105,149)(89,92,106,151)(90,129,107,141), (1,131,6,136)(2,137,7,132)(3,133,8,138)(4,139,9,134)(5,135,10,140)(11,36,16,31)(12,32,17,37)(13,38,18,33)(14,34,19,39)(15,40,20,35)(21,26,88,83)(22,84,89,27)(23,28,90,85)(24,86,81,29)(25,30,82,87)(41,74,54,62)(42,63,55,75)(43,76,56,64)(44,65,57,77)(45,78,58,66)(46,67,59,79)(47,80,60,68)(48,69,51,71)(49,72,52,70)(50,61,53,73)(91,122,149,155)(92,156,150,123)(93,124,141,157)(94,158,142,125)(95,126,143,159)(96,160,144,127)(97,128,145,151)(98,152,146,129)(99,130,147,153)(100,154,148,121)(101,106,118,113)(102,114,119,107)(103,108,120,115)(104,116,111,109)(105,110,112,117), (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), (1,86,131,81,6,29,136,24)(2,90,137,85,7,23,132,28)(3,84,133,89,8,27,138,22)(4,88,139,83,9,21,134,26)(5,82,135,87,10,25,140,30)(11,103,36,108,16,120,31,115)(12,107,32,102,17,114,37,119)(13,101,38,106,18,118,33,113)(14,105,34,110,19,112,39,117)(15,109,40,104,20,116,35,111)(41,160,74,144,54,127,62,96)(42,143,63,159,55,95,75,126)(43,158,76,142,56,125,64,94)(44,141,65,157,57,93,77,124)(45,156,78,150,58,123,66,92)(46,149,67,155,59,91,79,122)(47,154,80,148,60,121,68,100)(48,147,69,153,51,99,71,130)(49,152,72,146,52,129,70,98)(50,145,61,151,53,97,73,128) );
G=PermutationGroup([[(1,43,11,51),(2,49,12,57),(3,45,13,53),(4,41,14,59),(5,47,15,55),(6,56,16,48),(7,52,17,44),(8,58,18,50),(9,54,19,46),(10,60,20,42),(21,160,112,91),(22,150,113,128),(23,152,114,93),(24,142,115,130),(25,154,116,95),(26,144,117,122),(27,156,118,97),(28,146,119,124),(29,158,120,99),(30,148,111,126),(31,69,136,64),(32,77,137,72),(33,61,138,66),(34,79,139,74),(35,63,140,68),(36,71,131,76),(37,65,132,70),(38,73,133,78),(39,67,134,62),(40,75,135,80),(81,94,108,153),(82,121,109,143),(83,96,110,155),(84,123,101,145),(85,98,102,157),(86,125,103,147),(87,100,104,159),(88,127,105,149),(89,92,106,151),(90,129,107,141)], [(1,131,6,136),(2,137,7,132),(3,133,8,138),(4,139,9,134),(5,135,10,140),(11,36,16,31),(12,32,17,37),(13,38,18,33),(14,34,19,39),(15,40,20,35),(21,26,88,83),(22,84,89,27),(23,28,90,85),(24,86,81,29),(25,30,82,87),(41,74,54,62),(42,63,55,75),(43,76,56,64),(44,65,57,77),(45,78,58,66),(46,67,59,79),(47,80,60,68),(48,69,51,71),(49,72,52,70),(50,61,53,73),(91,122,149,155),(92,156,150,123),(93,124,141,157),(94,158,142,125),(95,126,143,159),(96,160,144,127),(97,128,145,151),(98,152,146,129),(99,130,147,153),(100,154,148,121),(101,106,118,113),(102,114,119,107),(103,108,120,115),(104,116,111,109),(105,110,112,117)], [(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)], [(1,86,131,81,6,29,136,24),(2,90,137,85,7,23,132,28),(3,84,133,89,8,27,138,22),(4,88,139,83,9,21,134,26),(5,82,135,87,10,25,140,30),(11,103,36,108,16,120,31,115),(12,107,32,102,17,114,37,119),(13,101,38,106,18,118,33,113),(14,105,34,110,19,112,39,117),(15,109,40,104,20,116,35,111),(41,160,74,144,54,127,62,96),(42,143,63,159,55,95,75,126),(43,158,76,142,56,125,64,94),(44,141,65,157,57,93,77,124),(45,156,78,150,58,123,66,92),(46,149,67,155,59,91,79,122),(47,154,80,148,60,121,68,100),(48,147,69,153,51,99,71,130),(49,152,72,146,52,129,70,98),(50,145,61,151,53,97,73,128)]])
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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | D4.9D10 |
| kernel | C42.65D10 | C42.D5 | C20⋊2Q8 | C2×D4.D5 | C2×C5⋊Q16 | C5×C4.4D4 | C5⋊2C8 | C2×C20 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 |
Matrix representation of C42.65D10 ►in GL8(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 9 | 6 | 25 |
| 0 | 0 | 0 | 0 | 38 | 11 | 40 | 35 |
| 0 | 0 | 0 | 0 | 25 | 12 | 6 | 32 |
| 0 | 0 | 0 | 0 | 22 | 34 | 1 | 19 |
| 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 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 37 | 1 | 24 | 4 |
| 0 | 0 | 0 | 0 | 38 | 11 | 40 | 0 |
| 0 | 0 | 0 | 0 | 19 | 36 | 34 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 26 | 12 | 7 |
| 0 | 0 | 0 | 0 | 31 | 25 | 13 | 35 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 15 | 28 |
| 0 | 0 | 0 | 0 | 4 | 0 | 31 | 37 |
| 0 | 0 | 0 | 0 | 2 | 20 | 9 | 36 |
| 0 | 0 | 0 | 0 | 9 | 1 | 7 | 19 |
G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,5,38,25,22,0,0,0,0,9,11,12,34,0,0,0,0,6,40,6,1,0,0,0,0,25,35,32,19],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,37,38,19,0,0,0,0,0,1,11,36,0,0,0,0,34,24,40,34,0,0,0,0,1,4,0,0],[18,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,1,7,26,31,0,0,0,0,34,34,26,25,0,0,0,0,0,0,12,13,0,0,0,0,0,0,7,35],[0,0,18,0,0,0,0,0,0,0,0,23,0,0,0,0,16,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,13,4,2,9,0,0,0,0,0,0,20,1,0,0,0,0,15,31,9,7,0,0,0,0,28,37,36,19] >;
C42.65D10 in GAP, Magma, Sage, TeX
C_4^2._{65}D_{10} % in TeX
G:=Group("C4^2.65D10"); // GroupNames label
G:=SmallGroup(320,687);
// by ID
G=gap.SmallGroup(320,687);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,555,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations