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), (C4×C20).162C22, (C2×C20).588C23, (C2×C10).109C24, 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
Subgroups: 838 in 238 conjugacy classes, 99 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], Dic5 [×6], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×3], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20.6Q8, C4×D20, Dic5⋊4D4 [×2], D10.12D4 [×2], C4⋊C4⋊D5 [×2], C2×C4⋊Dic5, C23.21D10, C20⋊7D4 [×2], Dic5⋊D4 [×2], D4×C20, C42.119D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.47C24, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊8D10, C42.119D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 110 63 116)(2 144 64 82)(3 102 65 118)(4 146 66 84)(5 104 67 120)(6 148 68 86)(7 106 69 112)(8 150 70 88)(9 108 61 114)(10 142 62 90)(11 41 95 129)(12 159 96 135)(13 43 97 121)(14 151 98 137)(15 45 99 123)(16 153 100 139)(17 47 91 125)(18 155 92 131)(19 49 93 127)(20 157 94 133)(21 158 78 134)(22 42 79 130)(23 160 80 136)(24 44 71 122)(25 152 72 138)(26 46 73 124)(27 154 74 140)(28 48 75 126)(29 156 76 132)(30 50 77 128)(31 117 60 101)(32 83 51 145)(33 119 52 103)(34 85 53 147)(35 111 54 105)(36 87 55 149)(37 113 56 107)(38 89 57 141)(39 115 58 109)(40 81 59 143)
(1 100 40 26)(2 91 31 27)(3 92 32 28)(4 93 33 29)(5 94 34 30)(6 95 35 21)(7 96 36 22)(8 97 37 23)(9 98 38 24)(10 99 39 25)(11 54 78 68)(12 55 79 69)(13 56 80 70)(14 57 71 61)(15 58 72 62)(16 59 73 63)(17 60 74 64)(18 51 75 65)(19 52 76 66)(20 53 77 67)(41 105 134 86)(42 106 135 87)(43 107 136 88)(44 108 137 89)(45 109 138 90)(46 110 139 81)(47 101 140 82)(48 102 131 83)(49 103 132 84)(50 104 133 85)(111 158 148 129)(112 159 149 130)(113 160 150 121)(114 151 141 122)(115 152 142 123)(116 153 143 124)(117 154 144 125)(118 155 145 126)(119 156 146 127)(120 157 147 128)
(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 30 95 77)(12 76 96 29)(13 28 97 75)(14 74 98 27)(15 26 99 73)(16 72 100 25)(17 24 91 71)(18 80 92 23)(19 22 93 79)(20 78 94 21)(31 57 60 38)(32 37 51 56)(33 55 52 36)(34 35 53 54)(39 59 58 40)(41 157 129 133)(42 132 130 156)(43 155 121 131)(44 140 122 154)(45 153 123 139)(46 138 124 152)(47 151 125 137)(48 136 126 160)(49 159 127 135)(50 134 128 158)(81 90 143 142)(82 141 144 89)(83 88 145 150)(84 149 146 87)(85 86 147 148)(101 114 117 108)(102 107 118 113)(103 112 119 106)(104 105 120 111)(109 116 115 110)
G:=sub<Sym(160)| (1,110,63,116)(2,144,64,82)(3,102,65,118)(4,146,66,84)(5,104,67,120)(6,148,68,86)(7,106,69,112)(8,150,70,88)(9,108,61,114)(10,142,62,90)(11,41,95,129)(12,159,96,135)(13,43,97,121)(14,151,98,137)(15,45,99,123)(16,153,100,139)(17,47,91,125)(18,155,92,131)(19,49,93,127)(20,157,94,133)(21,158,78,134)(22,42,79,130)(23,160,80,136)(24,44,71,122)(25,152,72,138)(26,46,73,124)(27,154,74,140)(28,48,75,126)(29,156,76,132)(30,50,77,128)(31,117,60,101)(32,83,51,145)(33,119,52,103)(34,85,53,147)(35,111,54,105)(36,87,55,149)(37,113,56,107)(38,89,57,141)(39,115,58,109)(40,81,59,143), (1,100,40,26)(2,91,31,27)(3,92,32,28)(4,93,33,29)(5,94,34,30)(6,95,35,21)(7,96,36,22)(8,97,37,23)(9,98,38,24)(10,99,39,25)(11,54,78,68)(12,55,79,69)(13,56,80,70)(14,57,71,61)(15,58,72,62)(16,59,73,63)(17,60,74,64)(18,51,75,65)(19,52,76,66)(20,53,77,67)(41,105,134,86)(42,106,135,87)(43,107,136,88)(44,108,137,89)(45,109,138,90)(46,110,139,81)(47,101,140,82)(48,102,131,83)(49,103,132,84)(50,104,133,85)(111,158,148,129)(112,159,149,130)(113,160,150,121)(114,151,141,122)(115,152,142,123)(116,153,143,124)(117,154,144,125)(118,155,145,126)(119,156,146,127)(120,157,147,128), (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)(11,30,95,77)(12,76,96,29)(13,28,97,75)(14,74,98,27)(15,26,99,73)(16,72,100,25)(17,24,91,71)(18,80,92,23)(19,22,93,79)(20,78,94,21)(31,57,60,38)(32,37,51,56)(33,55,52,36)(34,35,53,54)(39,59,58,40)(41,157,129,133)(42,132,130,156)(43,155,121,131)(44,140,122,154)(45,153,123,139)(46,138,124,152)(47,151,125,137)(48,136,126,160)(49,159,127,135)(50,134,128,158)(81,90,143,142)(82,141,144,89)(83,88,145,150)(84,149,146,87)(85,86,147,148)(101,114,117,108)(102,107,118,113)(103,112,119,106)(104,105,120,111)(109,116,115,110)>;
G:=Group( (1,110,63,116)(2,144,64,82)(3,102,65,118)(4,146,66,84)(5,104,67,120)(6,148,68,86)(7,106,69,112)(8,150,70,88)(9,108,61,114)(10,142,62,90)(11,41,95,129)(12,159,96,135)(13,43,97,121)(14,151,98,137)(15,45,99,123)(16,153,100,139)(17,47,91,125)(18,155,92,131)(19,49,93,127)(20,157,94,133)(21,158,78,134)(22,42,79,130)(23,160,80,136)(24,44,71,122)(25,152,72,138)(26,46,73,124)(27,154,74,140)(28,48,75,126)(29,156,76,132)(30,50,77,128)(31,117,60,101)(32,83,51,145)(33,119,52,103)(34,85,53,147)(35,111,54,105)(36,87,55,149)(37,113,56,107)(38,89,57,141)(39,115,58,109)(40,81,59,143), (1,100,40,26)(2,91,31,27)(3,92,32,28)(4,93,33,29)(5,94,34,30)(6,95,35,21)(7,96,36,22)(8,97,37,23)(9,98,38,24)(10,99,39,25)(11,54,78,68)(12,55,79,69)(13,56,80,70)(14,57,71,61)(15,58,72,62)(16,59,73,63)(17,60,74,64)(18,51,75,65)(19,52,76,66)(20,53,77,67)(41,105,134,86)(42,106,135,87)(43,107,136,88)(44,108,137,89)(45,109,138,90)(46,110,139,81)(47,101,140,82)(48,102,131,83)(49,103,132,84)(50,104,133,85)(111,158,148,129)(112,159,149,130)(113,160,150,121)(114,151,141,122)(115,152,142,123)(116,153,143,124)(117,154,144,125)(118,155,145,126)(119,156,146,127)(120,157,147,128), (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)(11,30,95,77)(12,76,96,29)(13,28,97,75)(14,74,98,27)(15,26,99,73)(16,72,100,25)(17,24,91,71)(18,80,92,23)(19,22,93,79)(20,78,94,21)(31,57,60,38)(32,37,51,56)(33,55,52,36)(34,35,53,54)(39,59,58,40)(41,157,129,133)(42,132,130,156)(43,155,121,131)(44,140,122,154)(45,153,123,139)(46,138,124,152)(47,151,125,137)(48,136,126,160)(49,159,127,135)(50,134,128,158)(81,90,143,142)(82,141,144,89)(83,88,145,150)(84,149,146,87)(85,86,147,148)(101,114,117,108)(102,107,118,113)(103,112,119,106)(104,105,120,111)(109,116,115,110) );
G=PermutationGroup([(1,110,63,116),(2,144,64,82),(3,102,65,118),(4,146,66,84),(5,104,67,120),(6,148,68,86),(7,106,69,112),(8,150,70,88),(9,108,61,114),(10,142,62,90),(11,41,95,129),(12,159,96,135),(13,43,97,121),(14,151,98,137),(15,45,99,123),(16,153,100,139),(17,47,91,125),(18,155,92,131),(19,49,93,127),(20,157,94,133),(21,158,78,134),(22,42,79,130),(23,160,80,136),(24,44,71,122),(25,152,72,138),(26,46,73,124),(27,154,74,140),(28,48,75,126),(29,156,76,132),(30,50,77,128),(31,117,60,101),(32,83,51,145),(33,119,52,103),(34,85,53,147),(35,111,54,105),(36,87,55,149),(37,113,56,107),(38,89,57,141),(39,115,58,109),(40,81,59,143)], [(1,100,40,26),(2,91,31,27),(3,92,32,28),(4,93,33,29),(5,94,34,30),(6,95,35,21),(7,96,36,22),(8,97,37,23),(9,98,38,24),(10,99,39,25),(11,54,78,68),(12,55,79,69),(13,56,80,70),(14,57,71,61),(15,58,72,62),(16,59,73,63),(17,60,74,64),(18,51,75,65),(19,52,76,66),(20,53,77,67),(41,105,134,86),(42,106,135,87),(43,107,136,88),(44,108,137,89),(45,109,138,90),(46,110,139,81),(47,101,140,82),(48,102,131,83),(49,103,132,84),(50,104,133,85),(111,158,148,129),(112,159,149,130),(113,160,150,121),(114,151,141,122),(115,152,142,123),(116,153,143,124),(117,154,144,125),(118,155,145,126),(119,156,146,127),(120,157,147,128)], [(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),(11,30,95,77),(12,76,96,29),(13,28,97,75),(14,74,98,27),(15,26,99,73),(16,72,100,25),(17,24,91,71),(18,80,92,23),(19,22,93,79),(20,78,94,21),(31,57,60,38),(32,37,51,56),(33,55,52,36),(34,35,53,54),(39,59,58,40),(41,157,129,133),(42,132,130,156),(43,155,121,131),(44,140,122,154),(45,153,123,139),(46,138,124,152),(47,151,125,137),(48,136,126,160),(49,159,127,135),(50,134,128,158),(81,90,143,142),(82,141,144,89),(83,88,145,150),(84,149,146,87),(85,86,147,148),(101,114,117,108),(102,107,118,113),(103,112,119,106),(104,105,120,111),(109,116,115,110)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀