metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.116D10, C10.1052+ (1+4), (C4×D4)⋊24D5, (D4×C20)⋊26C2, (C4×D20)⋊33C2, C20⋊7D4⋊12C2, C4⋊2D20⋊16C2, C20⋊D4⋊10C2, C20⋊2D4⋊10C2, C4⋊C4.287D10, D10⋊D4⋊11C2, (C2×D4).223D10, C4.46(C4○D20), C42⋊D5⋊15C2, C4.Dic10⋊16C2, D10.12D4⋊9C2, C20.113(C4○D4), (C2×C20).164C23, (C4×C20).160C22, (C2×C10).106C24, C22⋊C4.118D10, (C22×C4).214D10, C2.18(D4⋊8D10), C2.24(D4⋊6D10), (D4×C10).265C22, (C2×D20).268C22, C23.23D10⋊4C2, C4⋊Dic5.364C22, (C22×C20).83C22, (C2×Dic5).47C23, (C4×Dic5).86C22, C10.D4.7C22, (C22×D5).40C23, C23.103(C22×D5), C22.131(C23×D5), C23.D5.16C22, D10⋊C4.88C22, (C22×C10).176C23, C5⋊2(C22.34C24), C10.48(C2×C4○D4), C2.55(C2×C4○D20), (C2×C4×D5).254C22, (C5×C4⋊C4).334C22, (C2×C4).581(C22×D5), (C2×C5⋊D4).19C22, (C5×C22⋊C4).129C22, SmallGroup(320,1234)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×12], C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×9], Dic5 [×5], C20 [×2], C20 [×4], D10 [×9], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×D4, C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×D5 [×2], C22×C10 [×2], C22.34C24, C4×Dic5, C10.D4 [×2], C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C2×C5⋊D4 [×6], C22×C20 [×2], D4×C10, C42⋊D5, C4×D20, D10.12D4 [×2], D10⋊D4 [×2], C4.Dic10, C4⋊2D20, C23.23D10 [×2], C20⋊7D4 [×2], C20⋊2D4, C20⋊D4, D4×C20, C42.116D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.34C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊6D10, D4⋊8D10, C42.116D10
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=a2b-1, dcd-1=a2c-1 >
►On 160 points
Generators in S160
(1 110 63 114)(2 144 64 90)(3 102 65 116)(4 146 66 82)(5 104 67 118)(6 148 68 84)(7 106 69 120)(8 150 70 86)(9 108 61 112)(10 142 62 88)(11 43 97 129)(12 151 98 135)(13 45 99 121)(14 153 100 137)(15 47 91 123)(16 155 92 139)(17 49 93 125)(18 157 94 131)(19 41 95 127)(20 159 96 133)(21 160 80 134)(22 44 71 130)(23 152 72 136)(24 46 73 122)(25 154 74 138)(26 48 75 124)(27 156 76 140)(28 50 77 126)(29 158 78 132)(30 42 79 128)(31 115 58 101)(32 81 59 145)(33 117 60 103)(34 83 51 147)(35 119 52 105)(36 85 53 149)(37 111 54 107)(38 87 55 141)(39 113 56 109)(40 89 57 143)
(1 100 40 24)(2 91 31 25)(3 92 32 26)(4 93 33 27)(5 94 34 28)(6 95 35 29)(7 96 36 30)(8 97 37 21)(9 98 38 22)(10 99 39 23)(11 54 80 70)(12 55 71 61)(13 56 72 62)(14 57 73 63)(15 58 74 64)(16 59 75 65)(17 60 76 66)(18 51 77 67)(19 52 78 68)(20 53 79 69)(41 105 132 84)(42 106 133 85)(43 107 134 86)(44 108 135 87)(45 109 136 88)(46 110 137 89)(47 101 138 90)(48 102 139 81)(49 103 140 82)(50 104 131 83)(111 160 150 129)(112 151 141 130)(113 152 142 121)(114 153 143 122)(115 154 144 123)(116 155 145 124)(117 156 146 125)(118 157 147 126)(119 158 148 127)(120 159 149 128)
(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)
(1 121 63 45)(2 44 64 130)(3 129 65 43)(4 42 66 128)(5 127 67 41)(6 50 68 126)(7 125 69 49)(8 48 70 124)(9 123 61 47)(10 46 62 122)(11 81 97 145)(12 144 98 90)(13 89 99 143)(14 142 100 88)(15 87 91 141)(16 150 92 86)(17 85 93 149)(18 148 94 84)(19 83 95 147)(20 146 96 82)(21 116 80 102)(22 101 71 115)(23 114 72 110)(24 109 73 113)(25 112 74 108)(26 107 75 111)(27 120 76 106)(28 105 77 119)(29 118 78 104)(30 103 79 117)(31 135 58 151)(32 160 59 134)(33 133 60 159)(34 158 51 132)(35 131 52 157)(36 156 53 140)(37 139 54 155)(38 154 55 138)(39 137 56 153)(40 152 57 136)
G:=sub<Sym(160)| (1,110,63,114)(2,144,64,90)(3,102,65,116)(4,146,66,82)(5,104,67,118)(6,148,68,84)(7,106,69,120)(8,150,70,86)(9,108,61,112)(10,142,62,88)(11,43,97,129)(12,151,98,135)(13,45,99,121)(14,153,100,137)(15,47,91,123)(16,155,92,139)(17,49,93,125)(18,157,94,131)(19,41,95,127)(20,159,96,133)(21,160,80,134)(22,44,71,130)(23,152,72,136)(24,46,73,122)(25,154,74,138)(26,48,75,124)(27,156,76,140)(28,50,77,126)(29,158,78,132)(30,42,79,128)(31,115,58,101)(32,81,59,145)(33,117,60,103)(34,83,51,147)(35,119,52,105)(36,85,53,149)(37,111,54,107)(38,87,55,141)(39,113,56,109)(40,89,57,143), (1,100,40,24)(2,91,31,25)(3,92,32,26)(4,93,33,27)(5,94,34,28)(6,95,35,29)(7,96,36,30)(8,97,37,21)(9,98,38,22)(10,99,39,23)(11,54,80,70)(12,55,71,61)(13,56,72,62)(14,57,73,63)(15,58,74,64)(16,59,75,65)(17,60,76,66)(18,51,77,67)(19,52,78,68)(20,53,79,69)(41,105,132,84)(42,106,133,85)(43,107,134,86)(44,108,135,87)(45,109,136,88)(46,110,137,89)(47,101,138,90)(48,102,139,81)(49,103,140,82)(50,104,131,83)(111,160,150,129)(112,151,141,130)(113,152,142,121)(114,153,143,122)(115,154,144,123)(116,155,145,124)(117,156,146,125)(118,157,147,126)(119,158,148,127)(120,159,149,128), (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), (1,121,63,45)(2,44,64,130)(3,129,65,43)(4,42,66,128)(5,127,67,41)(6,50,68,126)(7,125,69,49)(8,48,70,124)(9,123,61,47)(10,46,62,122)(11,81,97,145)(12,144,98,90)(13,89,99,143)(14,142,100,88)(15,87,91,141)(16,150,92,86)(17,85,93,149)(18,148,94,84)(19,83,95,147)(20,146,96,82)(21,116,80,102)(22,101,71,115)(23,114,72,110)(24,109,73,113)(25,112,74,108)(26,107,75,111)(27,120,76,106)(28,105,77,119)(29,118,78,104)(30,103,79,117)(31,135,58,151)(32,160,59,134)(33,133,60,159)(34,158,51,132)(35,131,52,157)(36,156,53,140)(37,139,54,155)(38,154,55,138)(39,137,56,153)(40,152,57,136)>;
G:=Group( (1,110,63,114)(2,144,64,90)(3,102,65,116)(4,146,66,82)(5,104,67,118)(6,148,68,84)(7,106,69,120)(8,150,70,86)(9,108,61,112)(10,142,62,88)(11,43,97,129)(12,151,98,135)(13,45,99,121)(14,153,100,137)(15,47,91,123)(16,155,92,139)(17,49,93,125)(18,157,94,131)(19,41,95,127)(20,159,96,133)(21,160,80,134)(22,44,71,130)(23,152,72,136)(24,46,73,122)(25,154,74,138)(26,48,75,124)(27,156,76,140)(28,50,77,126)(29,158,78,132)(30,42,79,128)(31,115,58,101)(32,81,59,145)(33,117,60,103)(34,83,51,147)(35,119,52,105)(36,85,53,149)(37,111,54,107)(38,87,55,141)(39,113,56,109)(40,89,57,143), (1,100,40,24)(2,91,31,25)(3,92,32,26)(4,93,33,27)(5,94,34,28)(6,95,35,29)(7,96,36,30)(8,97,37,21)(9,98,38,22)(10,99,39,23)(11,54,80,70)(12,55,71,61)(13,56,72,62)(14,57,73,63)(15,58,74,64)(16,59,75,65)(17,60,76,66)(18,51,77,67)(19,52,78,68)(20,53,79,69)(41,105,132,84)(42,106,133,85)(43,107,134,86)(44,108,135,87)(45,109,136,88)(46,110,137,89)(47,101,138,90)(48,102,139,81)(49,103,140,82)(50,104,131,83)(111,160,150,129)(112,151,141,130)(113,152,142,121)(114,153,143,122)(115,154,144,123)(116,155,145,124)(117,156,146,125)(118,157,147,126)(119,158,148,127)(120,159,149,128), (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), (1,121,63,45)(2,44,64,130)(3,129,65,43)(4,42,66,128)(5,127,67,41)(6,50,68,126)(7,125,69,49)(8,48,70,124)(9,123,61,47)(10,46,62,122)(11,81,97,145)(12,144,98,90)(13,89,99,143)(14,142,100,88)(15,87,91,141)(16,150,92,86)(17,85,93,149)(18,148,94,84)(19,83,95,147)(20,146,96,82)(21,116,80,102)(22,101,71,115)(23,114,72,110)(24,109,73,113)(25,112,74,108)(26,107,75,111)(27,120,76,106)(28,105,77,119)(29,118,78,104)(30,103,79,117)(31,135,58,151)(32,160,59,134)(33,133,60,159)(34,158,51,132)(35,131,52,157)(36,156,53,140)(37,139,54,155)(38,154,55,138)(39,137,56,153)(40,152,57,136) );
G=PermutationGroup([(1,110,63,114),(2,144,64,90),(3,102,65,116),(4,146,66,82),(5,104,67,118),(6,148,68,84),(7,106,69,120),(8,150,70,86),(9,108,61,112),(10,142,62,88),(11,43,97,129),(12,151,98,135),(13,45,99,121),(14,153,100,137),(15,47,91,123),(16,155,92,139),(17,49,93,125),(18,157,94,131),(19,41,95,127),(20,159,96,133),(21,160,80,134),(22,44,71,130),(23,152,72,136),(24,46,73,122),(25,154,74,138),(26,48,75,124),(27,156,76,140),(28,50,77,126),(29,158,78,132),(30,42,79,128),(31,115,58,101),(32,81,59,145),(33,117,60,103),(34,83,51,147),(35,119,52,105),(36,85,53,149),(37,111,54,107),(38,87,55,141),(39,113,56,109),(40,89,57,143)], [(1,100,40,24),(2,91,31,25),(3,92,32,26),(4,93,33,27),(5,94,34,28),(6,95,35,29),(7,96,36,30),(8,97,37,21),(9,98,38,22),(10,99,39,23),(11,54,80,70),(12,55,71,61),(13,56,72,62),(14,57,73,63),(15,58,74,64),(16,59,75,65),(17,60,76,66),(18,51,77,67),(19,52,78,68),(20,53,79,69),(41,105,132,84),(42,106,133,85),(43,107,134,86),(44,108,135,87),(45,109,136,88),(46,110,137,89),(47,101,138,90),(48,102,139,81),(49,103,140,82),(50,104,131,83),(111,160,150,129),(112,151,141,130),(113,152,142,121),(114,153,143,122),(115,154,144,123),(116,155,145,124),(117,156,146,125),(118,157,147,126),(119,158,148,127),(120,159,149,128)], [(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)], [(1,121,63,45),(2,44,64,130),(3,129,65,43),(4,42,66,128),(5,127,67,41),(6,50,68,126),(7,125,69,49),(8,48,70,124),(9,123,61,47),(10,46,62,122),(11,81,97,145),(12,144,98,90),(13,89,99,143),(14,142,100,88),(15,87,91,141),(16,150,92,86),(17,85,93,149),(18,148,94,84),(19,83,95,147),(20,146,96,82),(21,116,80,102),(22,101,71,115),(23,114,72,110),(24,109,73,113),(25,112,74,108),(26,107,75,111),(27,120,76,106),(28,105,77,119),(29,118,78,104),(30,103,79,117),(31,135,58,151),(32,160,59,134),(33,133,60,159),(34,158,51,132),(35,131,52,157),(36,156,53,140),(37,139,54,155),(38,154,55,138),(39,137,56,153),(40,152,57,136)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 9 | 0 |
| 0 | 0 | 0 | 30 | 0 | 9 |
| 0 | 0 | 32 | 0 | 11 | 0 |
| 0 | 0 | 0 | 32 | 0 | 11 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 17 | 39 | 0 | 0 | 0 | 0 |
| 21 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 17 | 39 | 0 | 0 | 0 | 0 |
| 22 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 11 | 33 | 23 |
| 0 | 0 | 27 | 27 | 8 | 8 |
| 0 | 0 | 8 | 18 | 27 | 30 |
| 0 | 0 | 33 | 33 | 14 | 14 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,0,32,0,0,0,0,30,0,32,0,0,9,0,11,0,0,0,0,9,0,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[17,21,0,0,0,0,39,24,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[17,22,0,0,0,0,39,24,0,0,0,0,0,0,14,27,8,33,0,0,11,27,18,33,0,0,33,8,27,14,0,0,23,8,30,14] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4M | 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 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4⋊6D10 | D4⋊8D10 |
| kernel | C42.116D10 | C42⋊D5 | C4×D20 | D10.12D4 | D10⋊D4 | C4.Dic10 | C4⋊2D20 | C23.23D10 | C20⋊7D4 | C20⋊2D4 | C20⋊D4 | D4×C20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{116}D_{10} % in TeX
G:=Group("C4^2.116D10"); // GroupNames label
G:=SmallGroup(320,1234);
// by ID
G=gap.SmallGroup(320,1234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,192,12550]);
// 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=a^2*b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀