metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.51D10, D4.D5⋊9C4, D4.6(C4×D5), (C4×D4).8D5, (D4×C20).9C2, C4⋊C4.246D10, C10.102(C4×D4), (C2×C20).256D4, Dic10⋊20(C2×C4), (C4×Dic10)⋊20C2, (C2×D4).193D10, C4.39(C4○D20), C20.53(C4○D4), C10.D8⋊32C2, C10.Q16⋊30C2, C5⋊6(SD16⋊C4), (C4×C20).89C22, C20.59(C22×C4), C42.D5⋊6C2, C10.88(C8⋊C22), (C2×C20).340C23, D4⋊Dic5.10C2, C2.3(D4.9D10), C2.4(D4.D10), (D4×C10).235C22, C4⋊Dic5.329C22, C10.108(C8.C22), (C2×Dic10).273C22, C4.24(C2×C4×D5), C5⋊2C8⋊9(C2×C4), C2.18(C4×C5⋊D4), (C5×D4).27(C2×C4), (C2×D4.D5).4C2, (C2×C10).471(C2×D4), C22.78(C2×C5⋊D4), (C2×C4).219(C5⋊D4), (C5×C4⋊C4).277C22, (C2×C5⋊2C8).95C22, (C2×C4).440(C22×D5), SmallGroup(320,645)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.51D10
G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 358 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, SD16⋊C4, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C42.D5, C10.D8, C10.Q16, D4⋊Dic5, C4×Dic10, C2×D4.D5, D4×C20, C42.51D10
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, D4.D10, D4.9D10, C42.51D10
►On 160 points
Generators in S160
(1 87 22 44)(2 83 23 50)(3 89 24 46)(4 85 25 42)(5 81 21 48)(6 92 20 56)(7 98 16 52)(8 94 17 58)(9 100 18 54)(10 96 19 60)(11 57 38 93)(12 53 39 99)(13 59 40 95)(14 55 36 91)(15 51 37 97)(26 86 34 43)(27 82 35 49)(28 88 31 45)(29 84 32 41)(30 90 33 47)(61 123 149 108)(62 131 150 111)(63 125 141 110)(64 133 142 113)(65 127 143 102)(66 135 144 115)(67 129 145 104)(68 137 146 117)(69 121 147 106)(70 139 148 119)(71 132 152 112)(72 126 153 101)(73 134 154 114)(74 128 155 103)(75 136 156 116)(76 130 157 105)(77 138 158 118)(78 122 159 107)(79 140 160 120)(80 124 151 109)
(1 58 27 53)(2 54 28 59)(3 60 29 55)(4 56 30 51)(5 52 26 57)(6 90 37 85)(7 86 38 81)(8 82 39 87)(9 88 40 83)(10 84 36 89)(11 48 16 43)(12 44 17 49)(13 50 18 45)(14 46 19 41)(15 42 20 47)(21 98 34 93)(22 94 35 99)(23 100 31 95)(24 96 32 91)(25 92 33 97)(61 74 79 66)(62 67 80 75)(63 76 71 68)(64 69 72 77)(65 78 73 70)(101 118 113 106)(102 107 114 119)(103 120 115 108)(104 109 116 111)(105 112 117 110)(121 126 138 133)(122 134 139 127)(123 128 140 135)(124 136 131 129)(125 130 132 137)(141 157 152 146)(142 147 153 158)(143 159 154 148)(144 149 155 160)(145 151 156 150)
(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 76 58 71 27 68 53 63)(2 80 54 75 28 62 59 67)(3 74 60 79 29 66 55 61)(4 78 56 73 30 70 51 65)(5 72 52 77 26 64 57 69)(6 127 90 122 37 134 85 139)(7 121 86 126 38 138 81 133)(8 125 82 130 39 132 87 137)(9 129 88 124 40 136 83 131)(10 123 84 128 36 140 89 135)(11 118 48 113 16 106 43 101)(12 112 44 117 17 110 49 105)(13 116 50 111 18 104 45 109)(14 120 46 115 19 108 41 103)(15 114 42 119 20 102 47 107)(21 153 98 158 34 142 93 147)(22 157 94 152 35 146 99 141)(23 151 100 156 31 150 95 145)(24 155 96 160 32 144 91 149)(25 159 92 154 33 148 97 143)
G:=sub<Sym(160)| (1,87,22,44)(2,83,23,50)(3,89,24,46)(4,85,25,42)(5,81,21,48)(6,92,20,56)(7,98,16,52)(8,94,17,58)(9,100,18,54)(10,96,19,60)(11,57,38,93)(12,53,39,99)(13,59,40,95)(14,55,36,91)(15,51,37,97)(26,86,34,43)(27,82,35,49)(28,88,31,45)(29,84,32,41)(30,90,33,47)(61,123,149,108)(62,131,150,111)(63,125,141,110)(64,133,142,113)(65,127,143,102)(66,135,144,115)(67,129,145,104)(68,137,146,117)(69,121,147,106)(70,139,148,119)(71,132,152,112)(72,126,153,101)(73,134,154,114)(74,128,155,103)(75,136,156,116)(76,130,157,105)(77,138,158,118)(78,122,159,107)(79,140,160,120)(80,124,151,109), (1,58,27,53)(2,54,28,59)(3,60,29,55)(4,56,30,51)(5,52,26,57)(6,90,37,85)(7,86,38,81)(8,82,39,87)(9,88,40,83)(10,84,36,89)(11,48,16,43)(12,44,17,49)(13,50,18,45)(14,46,19,41)(15,42,20,47)(21,98,34,93)(22,94,35,99)(23,100,31,95)(24,96,32,91)(25,92,33,97)(61,74,79,66)(62,67,80,75)(63,76,71,68)(64,69,72,77)(65,78,73,70)(101,118,113,106)(102,107,114,119)(103,120,115,108)(104,109,116,111)(105,112,117,110)(121,126,138,133)(122,134,139,127)(123,128,140,135)(124,136,131,129)(125,130,132,137)(141,157,152,146)(142,147,153,158)(143,159,154,148)(144,149,155,160)(145,151,156,150), (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), (1,76,58,71,27,68,53,63)(2,80,54,75,28,62,59,67)(3,74,60,79,29,66,55,61)(4,78,56,73,30,70,51,65)(5,72,52,77,26,64,57,69)(6,127,90,122,37,134,85,139)(7,121,86,126,38,138,81,133)(8,125,82,130,39,132,87,137)(9,129,88,124,40,136,83,131)(10,123,84,128,36,140,89,135)(11,118,48,113,16,106,43,101)(12,112,44,117,17,110,49,105)(13,116,50,111,18,104,45,109)(14,120,46,115,19,108,41,103)(15,114,42,119,20,102,47,107)(21,153,98,158,34,142,93,147)(22,157,94,152,35,146,99,141)(23,151,100,156,31,150,95,145)(24,155,96,160,32,144,91,149)(25,159,92,154,33,148,97,143)>;
G:=Group( (1,87,22,44)(2,83,23,50)(3,89,24,46)(4,85,25,42)(5,81,21,48)(6,92,20,56)(7,98,16,52)(8,94,17,58)(9,100,18,54)(10,96,19,60)(11,57,38,93)(12,53,39,99)(13,59,40,95)(14,55,36,91)(15,51,37,97)(26,86,34,43)(27,82,35,49)(28,88,31,45)(29,84,32,41)(30,90,33,47)(61,123,149,108)(62,131,150,111)(63,125,141,110)(64,133,142,113)(65,127,143,102)(66,135,144,115)(67,129,145,104)(68,137,146,117)(69,121,147,106)(70,139,148,119)(71,132,152,112)(72,126,153,101)(73,134,154,114)(74,128,155,103)(75,136,156,116)(76,130,157,105)(77,138,158,118)(78,122,159,107)(79,140,160,120)(80,124,151,109), (1,58,27,53)(2,54,28,59)(3,60,29,55)(4,56,30,51)(5,52,26,57)(6,90,37,85)(7,86,38,81)(8,82,39,87)(9,88,40,83)(10,84,36,89)(11,48,16,43)(12,44,17,49)(13,50,18,45)(14,46,19,41)(15,42,20,47)(21,98,34,93)(22,94,35,99)(23,100,31,95)(24,96,32,91)(25,92,33,97)(61,74,79,66)(62,67,80,75)(63,76,71,68)(64,69,72,77)(65,78,73,70)(101,118,113,106)(102,107,114,119)(103,120,115,108)(104,109,116,111)(105,112,117,110)(121,126,138,133)(122,134,139,127)(123,128,140,135)(124,136,131,129)(125,130,132,137)(141,157,152,146)(142,147,153,158)(143,159,154,148)(144,149,155,160)(145,151,156,150), (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), (1,76,58,71,27,68,53,63)(2,80,54,75,28,62,59,67)(3,74,60,79,29,66,55,61)(4,78,56,73,30,70,51,65)(5,72,52,77,26,64,57,69)(6,127,90,122,37,134,85,139)(7,121,86,126,38,138,81,133)(8,125,82,130,39,132,87,137)(9,129,88,124,40,136,83,131)(10,123,84,128,36,140,89,135)(11,118,48,113,16,106,43,101)(12,112,44,117,17,110,49,105)(13,116,50,111,18,104,45,109)(14,120,46,115,19,108,41,103)(15,114,42,119,20,102,47,107)(21,153,98,158,34,142,93,147)(22,157,94,152,35,146,99,141)(23,151,100,156,31,150,95,145)(24,155,96,160,32,144,91,149)(25,159,92,154,33,148,97,143) );
G=PermutationGroup([[(1,87,22,44),(2,83,23,50),(3,89,24,46),(4,85,25,42),(5,81,21,48),(6,92,20,56),(7,98,16,52),(8,94,17,58),(9,100,18,54),(10,96,19,60),(11,57,38,93),(12,53,39,99),(13,59,40,95),(14,55,36,91),(15,51,37,97),(26,86,34,43),(27,82,35,49),(28,88,31,45),(29,84,32,41),(30,90,33,47),(61,123,149,108),(62,131,150,111),(63,125,141,110),(64,133,142,113),(65,127,143,102),(66,135,144,115),(67,129,145,104),(68,137,146,117),(69,121,147,106),(70,139,148,119),(71,132,152,112),(72,126,153,101),(73,134,154,114),(74,128,155,103),(75,136,156,116),(76,130,157,105),(77,138,158,118),(78,122,159,107),(79,140,160,120),(80,124,151,109)], [(1,58,27,53),(2,54,28,59),(3,60,29,55),(4,56,30,51),(5,52,26,57),(6,90,37,85),(7,86,38,81),(8,82,39,87),(9,88,40,83),(10,84,36,89),(11,48,16,43),(12,44,17,49),(13,50,18,45),(14,46,19,41),(15,42,20,47),(21,98,34,93),(22,94,35,99),(23,100,31,95),(24,96,32,91),(25,92,33,97),(61,74,79,66),(62,67,80,75),(63,76,71,68),(64,69,72,77),(65,78,73,70),(101,118,113,106),(102,107,114,119),(103,120,115,108),(104,109,116,111),(105,112,117,110),(121,126,138,133),(122,134,139,127),(123,128,140,135),(124,136,131,129),(125,130,132,137),(141,157,152,146),(142,147,153,158),(143,159,154,148),(144,149,155,160),(145,151,156,150)], [(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)], [(1,76,58,71,27,68,53,63),(2,80,54,75,28,62,59,67),(3,74,60,79,29,66,55,61),(4,78,56,73,30,70,51,65),(5,72,52,77,26,64,57,69),(6,127,90,122,37,134,85,139),(7,121,86,126,38,138,81,133),(8,125,82,130,39,132,87,137),(9,129,88,124,40,136,83,131),(10,123,84,128,36,140,89,135),(11,118,48,113,16,106,43,101),(12,112,44,117,17,110,49,105),(13,116,50,111,18,104,45,109),(14,120,46,115,19,108,41,103),(15,114,42,119,20,102,47,107),(21,153,98,158,34,142,93,147),(22,157,94,152,35,146,99,141),(23,151,100,156,31,150,95,145),(24,155,96,160,32,144,91,149),(25,159,92,154,33,148,97,143)]])
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 | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | D4.D10 | D4.9D10 |
| kernel | C42.51D10 | C42.D5 | C10.D8 | C10.Q16 | D4⋊Dic5 | C4×Dic10 | C2×D4.D5 | D4×C20 | D4.D5 | C2×C20 | C4×D4 | C20 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | 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 C42.51D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 1 | 39 | 17 |
| 0 | 0 | 5 | 18 | 5 | 36 |
| 0 | 0 | 19 | 40 | 1 | 0 |
| 0 | 0 | 19 | 24 | 17 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 17 |
| 0 | 0 | 0 | 1 | 39 | 36 |
| 0 | 0 | 22 | 1 | 40 | 0 |
| 0 | 0 | 24 | 0 | 0 | 40 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 35 | 37 | 4 | 0 |
| 0 | 0 | 0 | 4 | 37 | 31 |
| 29 | 15 | 0 | 0 | 0 | 0 |
| 26 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 22 |
| 0 | 0 | 0 | 0 | 7 | 5 |
| 0 | 0 | 22 | 18 | 5 | 38 |
| 0 | 0 | 2 | 16 | 9 | 36 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,23,5,19,19,0,0,1,18,40,24,0,0,39,5,1,17,0,0,17,36,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,22,24,0,0,0,1,1,0,0,0,0,39,40,0,0,0,17,36,0,40],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,0,35,35,0,0,0,7,6,37,4,0,0,0,0,4,37,0,0,0,0,0,31],[29,26,0,0,0,0,15,12,0,0,0,0,0,0,0,0,22,2,0,0,0,0,18,16,0,0,26,7,5,9,0,0,22,5,38,36] >;
C42.51D10 in GAP, Magma, Sage, TeX
C_4^2._{51}D_{10} % in TeX
G:=Group("C4^2.51D10"); // GroupNames label
G:=SmallGroup(320,645);
// by ID
G=gap.SmallGroup(320,645);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,387,58,1684,851,102,12550]);
// 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=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