metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.156D10, C10.1342+ (1+4), C4⋊D20⋊15C2, C4⋊2D20⋊34C2, C4⋊C4.113D10, C42.C2⋊12D5, D20⋊8C4⋊38C2, C42⋊D5⋊22C2, C20.132(C4○D4), (C2×C10).242C24, (C2×C20).189C23, (C4×C20).201C22, C4.21(Q8⋊2D5), D10.13D4⋊36C2, C2.59(D4⋊8D10), (C2×D20).172C22, C22.263(C23×D5), C5⋊6(C22.34C24), (C2×Dic5).272C23, (C4×Dic5).155C22, (C22×D5).107C23, D10⋊C4.113C22, C10.D4.125C22, C10.119(C2×C4○D4), (C5×C42.C2)⋊15C2, C2.26(C2×Q8⋊2D5), (C2×C4×D5).141C22, (C5×C4⋊C4).197C22, (C2×C4).594(C22×D5), SmallGroup(320,1370)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1070 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4, C2×C4 [×6], C2×C4 [×9], D4 [×12], C23 [×5], D5 [×5], C10, C10 [×2], C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×5], C2×D4 [×10], Dic5 [×3], C20 [×2], C20 [×6], D10 [×15], C2×C10, C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D5 [×6], D20 [×12], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×6], C22×D5, C22×D5 [×4], C22.34C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×6], C2×C4×D5, C2×C4×D5 [×4], C2×D20 [×10], C42⋊D5, C4⋊D20, D20⋊8C4 [×2], D10.13D4 [×4], C4⋊2D20 [×6], C5×C42.C2, C42.156D10
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, Q8⋊2D5 [×2], C23×D5, C2×Q8⋊2D5, D4⋊8D10 [×2], C42.156D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 94 11 84)(2 137 12 127)(3 96 13 86)(4 139 14 129)(5 98 15 88)(6 121 16 131)(7 100 17 90)(8 123 18 133)(9 82 19 92)(10 125 20 135)(21 105 31 115)(22 68 32 78)(23 107 33 117)(24 70 34 80)(25 109 35 119)(26 72 36 62)(27 111 37 101)(28 74 38 64)(29 113 39 103)(30 76 40 66)(41 77 51 67)(42 116 52 106)(43 79 53 69)(44 118 54 108)(45 61 55 71)(46 120 56 110)(47 63 57 73)(48 102 58 112)(49 65 59 75)(50 104 60 114)(81 153 91 143)(83 155 93 145)(85 157 95 147)(87 159 97 149)(89 141 99 151)(122 152 132 142)(124 154 134 144)(126 156 136 146)(128 158 138 148)(130 160 140 150)
(1 110 156 72)(2 101 157 63)(3 112 158 74)(4 103 159 65)(5 114 160 76)(6 105 141 67)(7 116 142 78)(8 107 143 69)(9 118 144 80)(10 109 145 71)(11 120 146 62)(12 111 147 73)(13 102 148 64)(14 113 149 75)(15 104 150 66)(16 115 151 77)(17 106 152 68)(18 117 153 79)(19 108 154 70)(20 119 155 61)(21 89 51 131)(22 100 52 122)(23 91 53 133)(24 82 54 124)(25 93 55 135)(26 84 56 126)(27 95 57 137)(28 86 58 128)(29 97 59 139)(30 88 60 130)(31 99 41 121)(32 90 42 132)(33 81 43 123)(34 92 44 134)(35 83 45 125)(36 94 46 136)(37 85 47 127)(38 96 48 138)(39 87 49 129)(40 98 50 140)
(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 93 11 83)(2 82 12 92)(3 91 13 81)(4 100 14 90)(5 89 15 99)(6 98 16 88)(7 87 17 97)(8 96 18 86)(9 85 19 95)(10 94 20 84)(21 114 31 104)(22 103 32 113)(23 112 33 102)(24 101 34 111)(25 110 35 120)(26 119 36 109)(27 108 37 118)(28 117 38 107)(29 106 39 116)(30 115 40 105)(41 66 51 76)(42 75 52 65)(43 64 53 74)(44 73 54 63)(45 62 55 72)(46 71 56 61)(47 80 57 70)(48 69 58 79)(49 78 59 68)(50 67 60 77)(121 160 131 150)(122 149 132 159)(123 158 133 148)(124 147 134 157)(125 156 135 146)(126 145 136 155)(127 154 137 144)(128 143 138 153)(129 152 139 142)(130 141 140 151)
G:=sub<Sym(160)| (1,94,11,84)(2,137,12,127)(3,96,13,86)(4,139,14,129)(5,98,15,88)(6,121,16,131)(7,100,17,90)(8,123,18,133)(9,82,19,92)(10,125,20,135)(21,105,31,115)(22,68,32,78)(23,107,33,117)(24,70,34,80)(25,109,35,119)(26,72,36,62)(27,111,37,101)(28,74,38,64)(29,113,39,103)(30,76,40,66)(41,77,51,67)(42,116,52,106)(43,79,53,69)(44,118,54,108)(45,61,55,71)(46,120,56,110)(47,63,57,73)(48,102,58,112)(49,65,59,75)(50,104,60,114)(81,153,91,143)(83,155,93,145)(85,157,95,147)(87,159,97,149)(89,141,99,151)(122,152,132,142)(124,154,134,144)(126,156,136,146)(128,158,138,148)(130,160,140,150), (1,110,156,72)(2,101,157,63)(3,112,158,74)(4,103,159,65)(5,114,160,76)(6,105,141,67)(7,116,142,78)(8,107,143,69)(9,118,144,80)(10,109,145,71)(11,120,146,62)(12,111,147,73)(13,102,148,64)(14,113,149,75)(15,104,150,66)(16,115,151,77)(17,106,152,68)(18,117,153,79)(19,108,154,70)(20,119,155,61)(21,89,51,131)(22,100,52,122)(23,91,53,133)(24,82,54,124)(25,93,55,135)(26,84,56,126)(27,95,57,137)(28,86,58,128)(29,97,59,139)(30,88,60,130)(31,99,41,121)(32,90,42,132)(33,81,43,123)(34,92,44,134)(35,83,45,125)(36,94,46,136)(37,85,47,127)(38,96,48,138)(39,87,49,129)(40,98,50,140), (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), (1,93,11,83)(2,82,12,92)(3,91,13,81)(4,100,14,90)(5,89,15,99)(6,98,16,88)(7,87,17,97)(8,96,18,86)(9,85,19,95)(10,94,20,84)(21,114,31,104)(22,103,32,113)(23,112,33,102)(24,101,34,111)(25,110,35,120)(26,119,36,109)(27,108,37,118)(28,117,38,107)(29,106,39,116)(30,115,40,105)(41,66,51,76)(42,75,52,65)(43,64,53,74)(44,73,54,63)(45,62,55,72)(46,71,56,61)(47,80,57,70)(48,69,58,79)(49,78,59,68)(50,67,60,77)(121,160,131,150)(122,149,132,159)(123,158,133,148)(124,147,134,157)(125,156,135,146)(126,145,136,155)(127,154,137,144)(128,143,138,153)(129,152,139,142)(130,141,140,151)>;
G:=Group( (1,94,11,84)(2,137,12,127)(3,96,13,86)(4,139,14,129)(5,98,15,88)(6,121,16,131)(7,100,17,90)(8,123,18,133)(9,82,19,92)(10,125,20,135)(21,105,31,115)(22,68,32,78)(23,107,33,117)(24,70,34,80)(25,109,35,119)(26,72,36,62)(27,111,37,101)(28,74,38,64)(29,113,39,103)(30,76,40,66)(41,77,51,67)(42,116,52,106)(43,79,53,69)(44,118,54,108)(45,61,55,71)(46,120,56,110)(47,63,57,73)(48,102,58,112)(49,65,59,75)(50,104,60,114)(81,153,91,143)(83,155,93,145)(85,157,95,147)(87,159,97,149)(89,141,99,151)(122,152,132,142)(124,154,134,144)(126,156,136,146)(128,158,138,148)(130,160,140,150), (1,110,156,72)(2,101,157,63)(3,112,158,74)(4,103,159,65)(5,114,160,76)(6,105,141,67)(7,116,142,78)(8,107,143,69)(9,118,144,80)(10,109,145,71)(11,120,146,62)(12,111,147,73)(13,102,148,64)(14,113,149,75)(15,104,150,66)(16,115,151,77)(17,106,152,68)(18,117,153,79)(19,108,154,70)(20,119,155,61)(21,89,51,131)(22,100,52,122)(23,91,53,133)(24,82,54,124)(25,93,55,135)(26,84,56,126)(27,95,57,137)(28,86,58,128)(29,97,59,139)(30,88,60,130)(31,99,41,121)(32,90,42,132)(33,81,43,123)(34,92,44,134)(35,83,45,125)(36,94,46,136)(37,85,47,127)(38,96,48,138)(39,87,49,129)(40,98,50,140), (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), (1,93,11,83)(2,82,12,92)(3,91,13,81)(4,100,14,90)(5,89,15,99)(6,98,16,88)(7,87,17,97)(8,96,18,86)(9,85,19,95)(10,94,20,84)(21,114,31,104)(22,103,32,113)(23,112,33,102)(24,101,34,111)(25,110,35,120)(26,119,36,109)(27,108,37,118)(28,117,38,107)(29,106,39,116)(30,115,40,105)(41,66,51,76)(42,75,52,65)(43,64,53,74)(44,73,54,63)(45,62,55,72)(46,71,56,61)(47,80,57,70)(48,69,58,79)(49,78,59,68)(50,67,60,77)(121,160,131,150)(122,149,132,159)(123,158,133,148)(124,147,134,157)(125,156,135,146)(126,145,136,155)(127,154,137,144)(128,143,138,153)(129,152,139,142)(130,141,140,151) );
G=PermutationGroup([(1,94,11,84),(2,137,12,127),(3,96,13,86),(4,139,14,129),(5,98,15,88),(6,121,16,131),(7,100,17,90),(8,123,18,133),(9,82,19,92),(10,125,20,135),(21,105,31,115),(22,68,32,78),(23,107,33,117),(24,70,34,80),(25,109,35,119),(26,72,36,62),(27,111,37,101),(28,74,38,64),(29,113,39,103),(30,76,40,66),(41,77,51,67),(42,116,52,106),(43,79,53,69),(44,118,54,108),(45,61,55,71),(46,120,56,110),(47,63,57,73),(48,102,58,112),(49,65,59,75),(50,104,60,114),(81,153,91,143),(83,155,93,145),(85,157,95,147),(87,159,97,149),(89,141,99,151),(122,152,132,142),(124,154,134,144),(126,156,136,146),(128,158,138,148),(130,160,140,150)], [(1,110,156,72),(2,101,157,63),(3,112,158,74),(4,103,159,65),(5,114,160,76),(6,105,141,67),(7,116,142,78),(8,107,143,69),(9,118,144,80),(10,109,145,71),(11,120,146,62),(12,111,147,73),(13,102,148,64),(14,113,149,75),(15,104,150,66),(16,115,151,77),(17,106,152,68),(18,117,153,79),(19,108,154,70),(20,119,155,61),(21,89,51,131),(22,100,52,122),(23,91,53,133),(24,82,54,124),(25,93,55,135),(26,84,56,126),(27,95,57,137),(28,86,58,128),(29,97,59,139),(30,88,60,130),(31,99,41,121),(32,90,42,132),(33,81,43,123),(34,92,44,134),(35,83,45,125),(36,94,46,136),(37,85,47,127),(38,96,48,138),(39,87,49,129),(40,98,50,140)], [(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)], [(1,93,11,83),(2,82,12,92),(3,91,13,81),(4,100,14,90),(5,89,15,99),(6,98,16,88),(7,87,17,97),(8,96,18,86),(9,85,19,95),(10,94,20,84),(21,114,31,104),(22,103,32,113),(23,112,33,102),(24,101,34,111),(25,110,35,120),(26,119,36,109),(27,108,37,118),(28,117,38,107),(29,106,39,116),(30,115,40,105),(41,66,51,76),(42,75,52,65),(43,64,53,74),(44,73,54,63),(45,62,55,72),(46,71,56,61),(47,80,57,70),(48,69,58,79),(49,78,59,68),(50,67,60,77),(121,160,131,150),(122,149,132,159),(123,158,133,148),(124,147,134,157),(125,156,135,146),(126,145,136,155),(127,154,137,144),(128,143,138,153),(129,152,139,142),(130,141,140,151)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 | 21 | 21 |
| 0 | 0 | 0 | 0 | 19 | 30 | 40 | 38 |
| 0 | 0 | 0 | 0 | 6 | 1 | 2 | 28 |
| 0 | 0 | 0 | 0 | 39 | 40 | 13 | 39 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 2 |
| 40 | 1 | 8 | 8 | 0 | 0 | 0 | 0 |
| 40 | 33 | 33 | 34 | 0 | 0 | 0 | 0 |
| 8 | 8 | 1 | 40 | 0 | 0 | 0 | 0 |
| 33 | 34 | 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 | 18 | 20 |
| 0 | 0 | 0 | 0 | 22 | 26 | 21 | 21 |
| 33 | 33 | 40 | 1 | 0 | 0 | 0 | 0 |
| 7 | 8 | 8 | 1 | 0 | 0 | 0 | 0 |
| 1 | 40 | 33 | 33 | 0 | 0 | 0 | 0 |
| 33 | 40 | 7 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 9 | 1 | 6 |
| 0 | 0 | 0 | 0 | 19 | 19 | 40 | 34 |
| 0 | 0 | 0 | 0 | 14 | 12 | 32 | 0 |
| 0 | 0 | 0 | 0 | 39 | 39 | 13 | 9 |
G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,19,6,39,0,0,0,0,13,30,1,40,0,0,0,0,21,40,2,13,0,0,0,0,21,38,28,39],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,13,2],[40,40,8,33,0,0,0,0,1,33,8,34,0,0,0,0,8,33,1,1,0,0,0,0,8,34,40,8,0,0,0,0,0,0,0,0,3,37,13,22,0,0,0,0,21,40,4,26,0,0,0,0,0,0,18,21,0,0,0,0,0,0,20,21],[33,7,1,33,0,0,0,0,33,8,40,40,0,0,0,0,40,8,33,7,0,0,0,0,1,1,33,8,0,0,0,0,0,0,0,0,22,19,14,39,0,0,0,0,9,19,12,39,0,0,0,0,1,40,32,13,0,0,0,0,6,34,0,9] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | ··· | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ (1+4) | Q8⋊2D5 | D4⋊8D10 |
| kernel | C42.156D10 | C42⋊D5 | C4⋊D20 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | C5×C42.C2 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 6 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{156}D_{10} % in TeX
G:=Group("C4^2.156D10"); // GroupNames label
G:=SmallGroup(320,1370);
// by ID
G=gap.SmallGroup(320,1370);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,184,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀