metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.119D10, C10.1072+ 1+4, (C4×D4)⋊27D5, (D4×C20)⋊29C2, (C4×D20)⋊35C2, C20⋊7D4⋊13C2, C4⋊C4.289D10, (C2×D4).226D10, C20.6Q8⋊17C2, Dic5⋊D4⋊28C2, Dic5⋊4D4⋊48C2, C20.293(C4○D4), (C2×C10).109C24, (C2×C20).588C23, (C4×C20).162C22, C22⋊C4.121D10, C22.2(C4○D20), (C22×C4).216D10, D10.12D4⋊11C2, C2.20(D4⋊8D10), C4.119(D4⋊2D5), (D4×C10).310C22, (C2×D20).223C22, C4⋊Dic5.398C22, (C22×C20).84C22, (C2×Dic5).49C23, (C22×D5).43C23, C23.106(C22×D5), C22.134(C23×D5), C23.21D10⋊10C2, (C22×C10).179C23, C5⋊5(C22.47C24), (C4×Dic5).228C22, C10.D4.67C22, C23.D5.109C22, D10⋊C4.144C22, (C22×Dic5).101C22, C4⋊C4⋊D5⋊9C2, (C2×C4⋊Dic5)⋊26C2, C10.51(C2×C4○D4), C2.58(C2×C4○D20), C2.25(C2×D4⋊2D5), (C2×C4×D5).256C22, (C2×C10).19(C4○D4), (C5×C4⋊C4).337C22, (C2×C4).165(C22×D5), (C2×C5⋊D4).21C22, (C5×C22⋊C4).131C22, SmallGroup(320,1237)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.119D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 838 in 238 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.47C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C20.6Q8, C4×D20, Dic5⋊4D4, D10.12D4, C4⋊C4⋊D5, C2×C4⋊Dic5, C23.21D10, C20⋊7D4, Dic5⋊D4, D4×C20, C42.119D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊8D10, C42.119D10
►On 160 points
Generators in S160
(1 110 63 139)(2 144 64 117)(3 102 65 131)(4 146 66 119)(5 104 67 133)(6 148 68 111)(7 106 69 135)(8 150 70 113)(9 108 61 137)(10 142 62 115)(11 153 100 82)(12 47 91 123)(13 155 92 84)(14 49 93 125)(15 157 94 86)(16 41 95 127)(17 159 96 88)(18 43 97 129)(19 151 98 90)(20 45 99 121)(21 105 35 134)(22 149 36 112)(23 107 37 136)(24 141 38 114)(25 109 39 138)(26 143 40 116)(27 101 31 140)(28 145 32 118)(29 103 33 132)(30 147 34 120)(42 79 128 59)(44 71 130 51)(46 73 122 53)(48 75 124 55)(50 77 126 57)(52 152 72 81)(54 154 74 83)(56 156 76 85)(58 158 78 87)(60 160 80 89)
(1 100 40 53)(2 91 31 54)(3 92 32 55)(4 93 33 56)(5 94 34 57)(6 95 35 58)(7 96 36 59)(8 97 37 60)(9 98 38 51)(10 99 39 52)(11 26 73 63)(12 27 74 64)(13 28 75 65)(14 29 76 66)(15 30 77 67)(16 21 78 68)(17 22 79 69)(18 23 80 70)(19 24 71 61)(20 25 72 62)(41 105 87 111)(42 106 88 112)(43 107 89 113)(44 108 90 114)(45 109 81 115)(46 110 82 116)(47 101 83 117)(48 102 84 118)(49 103 85 119)(50 104 86 120)(121 138 152 142)(122 139 153 143)(123 140 154 144)(124 131 155 145)(125 132 156 146)(126 133 157 147)(127 134 158 148)(128 135 159 149)(129 136 160 150)(130 137 151 141)
(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 63 62)(2 61 64 9)(3 8 65 70)(4 69 66 7)(5 6 67 68)(11 72 100 52)(12 51 91 71)(13 80 92 60)(14 59 93 79)(15 78 94 58)(16 57 95 77)(17 76 96 56)(18 55 97 75)(19 74 98 54)(20 53 99 73)(21 34 35 30)(22 29 36 33)(23 32 37 28)(24 27 38 31)(25 40 39 26)(41 157 127 86)(42 85 128 156)(43 155 129 84)(44 83 130 154)(45 153 121 82)(46 81 122 152)(47 151 123 90)(48 89 124 160)(49 159 125 88)(50 87 126 158)(101 137 140 108)(102 107 131 136)(103 135 132 106)(104 105 133 134)(109 139 138 110)(111 147 148 120)(112 119 149 146)(113 145 150 118)(114 117 141 144)(115 143 142 116)
G:=sub<Sym(160)| (1,110,63,139)(2,144,64,117)(3,102,65,131)(4,146,66,119)(5,104,67,133)(6,148,68,111)(7,106,69,135)(8,150,70,113)(9,108,61,137)(10,142,62,115)(11,153,100,82)(12,47,91,123)(13,155,92,84)(14,49,93,125)(15,157,94,86)(16,41,95,127)(17,159,96,88)(18,43,97,129)(19,151,98,90)(20,45,99,121)(21,105,35,134)(22,149,36,112)(23,107,37,136)(24,141,38,114)(25,109,39,138)(26,143,40,116)(27,101,31,140)(28,145,32,118)(29,103,33,132)(30,147,34,120)(42,79,128,59)(44,71,130,51)(46,73,122,53)(48,75,124,55)(50,77,126,57)(52,152,72,81)(54,154,74,83)(56,156,76,85)(58,158,78,87)(60,160,80,89), (1,100,40,53)(2,91,31,54)(3,92,32,55)(4,93,33,56)(5,94,34,57)(6,95,35,58)(7,96,36,59)(8,97,37,60)(9,98,38,51)(10,99,39,52)(11,26,73,63)(12,27,74,64)(13,28,75,65)(14,29,76,66)(15,30,77,67)(16,21,78,68)(17,22,79,69)(18,23,80,70)(19,24,71,61)(20,25,72,62)(41,105,87,111)(42,106,88,112)(43,107,89,113)(44,108,90,114)(45,109,81,115)(46,110,82,116)(47,101,83,117)(48,102,84,118)(49,103,85,119)(50,104,86,120)(121,138,152,142)(122,139,153,143)(123,140,154,144)(124,131,155,145)(125,132,156,146)(126,133,157,147)(127,134,158,148)(128,135,159,149)(129,136,160,150)(130,137,151,141), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,72,100,52)(12,51,91,71)(13,80,92,60)(14,59,93,79)(15,78,94,58)(16,57,95,77)(17,76,96,56)(18,55,97,75)(19,74,98,54)(20,53,99,73)(21,34,35,30)(22,29,36,33)(23,32,37,28)(24,27,38,31)(25,40,39,26)(41,157,127,86)(42,85,128,156)(43,155,129,84)(44,83,130,154)(45,153,121,82)(46,81,122,152)(47,151,123,90)(48,89,124,160)(49,159,125,88)(50,87,126,158)(101,137,140,108)(102,107,131,136)(103,135,132,106)(104,105,133,134)(109,139,138,110)(111,147,148,120)(112,119,149,146)(113,145,150,118)(114,117,141,144)(115,143,142,116)>;
G:=Group( (1,110,63,139)(2,144,64,117)(3,102,65,131)(4,146,66,119)(5,104,67,133)(6,148,68,111)(7,106,69,135)(8,150,70,113)(9,108,61,137)(10,142,62,115)(11,153,100,82)(12,47,91,123)(13,155,92,84)(14,49,93,125)(15,157,94,86)(16,41,95,127)(17,159,96,88)(18,43,97,129)(19,151,98,90)(20,45,99,121)(21,105,35,134)(22,149,36,112)(23,107,37,136)(24,141,38,114)(25,109,39,138)(26,143,40,116)(27,101,31,140)(28,145,32,118)(29,103,33,132)(30,147,34,120)(42,79,128,59)(44,71,130,51)(46,73,122,53)(48,75,124,55)(50,77,126,57)(52,152,72,81)(54,154,74,83)(56,156,76,85)(58,158,78,87)(60,160,80,89), (1,100,40,53)(2,91,31,54)(3,92,32,55)(4,93,33,56)(5,94,34,57)(6,95,35,58)(7,96,36,59)(8,97,37,60)(9,98,38,51)(10,99,39,52)(11,26,73,63)(12,27,74,64)(13,28,75,65)(14,29,76,66)(15,30,77,67)(16,21,78,68)(17,22,79,69)(18,23,80,70)(19,24,71,61)(20,25,72,62)(41,105,87,111)(42,106,88,112)(43,107,89,113)(44,108,90,114)(45,109,81,115)(46,110,82,116)(47,101,83,117)(48,102,84,118)(49,103,85,119)(50,104,86,120)(121,138,152,142)(122,139,153,143)(123,140,154,144)(124,131,155,145)(125,132,156,146)(126,133,157,147)(127,134,158,148)(128,135,159,149)(129,136,160,150)(130,137,151,141), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,72,100,52)(12,51,91,71)(13,80,92,60)(14,59,93,79)(15,78,94,58)(16,57,95,77)(17,76,96,56)(18,55,97,75)(19,74,98,54)(20,53,99,73)(21,34,35,30)(22,29,36,33)(23,32,37,28)(24,27,38,31)(25,40,39,26)(41,157,127,86)(42,85,128,156)(43,155,129,84)(44,83,130,154)(45,153,121,82)(46,81,122,152)(47,151,123,90)(48,89,124,160)(49,159,125,88)(50,87,126,158)(101,137,140,108)(102,107,131,136)(103,135,132,106)(104,105,133,134)(109,139,138,110)(111,147,148,120)(112,119,149,146)(113,145,150,118)(114,117,141,144)(115,143,142,116) );
G=PermutationGroup([[(1,110,63,139),(2,144,64,117),(3,102,65,131),(4,146,66,119),(5,104,67,133),(6,148,68,111),(7,106,69,135),(8,150,70,113),(9,108,61,137),(10,142,62,115),(11,153,100,82),(12,47,91,123),(13,155,92,84),(14,49,93,125),(15,157,94,86),(16,41,95,127),(17,159,96,88),(18,43,97,129),(19,151,98,90),(20,45,99,121),(21,105,35,134),(22,149,36,112),(23,107,37,136),(24,141,38,114),(25,109,39,138),(26,143,40,116),(27,101,31,140),(28,145,32,118),(29,103,33,132),(30,147,34,120),(42,79,128,59),(44,71,130,51),(46,73,122,53),(48,75,124,55),(50,77,126,57),(52,152,72,81),(54,154,74,83),(56,156,76,85),(58,158,78,87),(60,160,80,89)], [(1,100,40,53),(2,91,31,54),(3,92,32,55),(4,93,33,56),(5,94,34,57),(6,95,35,58),(7,96,36,59),(8,97,37,60),(9,98,38,51),(10,99,39,52),(11,26,73,63),(12,27,74,64),(13,28,75,65),(14,29,76,66),(15,30,77,67),(16,21,78,68),(17,22,79,69),(18,23,80,70),(19,24,71,61),(20,25,72,62),(41,105,87,111),(42,106,88,112),(43,107,89,113),(44,108,90,114),(45,109,81,115),(46,110,82,116),(47,101,83,117),(48,102,84,118),(49,103,85,119),(50,104,86,120),(121,138,152,142),(122,139,153,143),(123,140,154,144),(124,131,155,145),(125,132,156,146),(126,133,157,147),(127,134,158,148),(128,135,159,149),(129,136,160,150),(130,137,151,141)], [(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)], [(1,10,63,62),(2,61,64,9),(3,8,65,70),(4,69,66,7),(5,6,67,68),(11,72,100,52),(12,51,91,71),(13,80,92,60),(14,59,93,79),(15,78,94,58),(16,57,95,77),(17,76,96,56),(18,55,97,75),(19,74,98,54),(20,53,99,73),(21,34,35,30),(22,29,36,33),(23,32,37,28),(24,27,38,31),(25,40,39,26),(41,157,127,86),(42,85,128,156),(43,155,129,84),(44,83,130,154),(45,153,121,82),(46,81,122,152),(47,151,123,90),(48,89,124,160),(49,159,125,88),(50,87,126,158),(101,137,140,108),(102,107,131,136),(103,135,132,106),(104,105,133,134),(109,139,138,110),(111,147,148,120),(112,119,149,146),(113,145,150,118),(114,117,141,144),(115,143,142,116)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊2D5 | D4⋊8D10 |
| kernel | C42.119D10 | C20.6Q8 | C4×D20 | Dic5⋊4D4 | D10.12D4 | C4⋊C4⋊D5 | C2×C4⋊Dic5 | C23.21D10 | C20⋊7D4 | Dic5⋊D4 | D4×C20 | C4×D4 | C20 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.119D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 20 | 0 | 0 |
| 21 | 18 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 21 | 20 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,1,0,0,0,0,1],[21,21,0,0,20,18,0,0,0,0,0,1,0,0,1,0],[21,23,0,0,20,20,0,0,0,0,0,1,0,0,40,0] >;
C42.119D10 in GAP, Magma, Sage, TeX
C_4^2._{119}D_{10} % in TeX
G:=Group("C4^2.119D10"); // GroupNames label
G:=SmallGroup(320,1237);
// by ID
G=gap.SmallGroup(320,1237);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations