metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.161D10, C10.1002- (1+4), C10.1392+ (1+4), (C4×D20)⋊15C2, C4⋊C4.118D10, C42⋊2C2⋊4D5, C20.6Q8⋊9C2, D10⋊Q8⋊43C2, D10⋊2Q8⋊41C2, D10⋊D4.4C2, (C4×C20).33C22, C22⋊C4.79D10, C4.Dic10⋊39C2, Dic5⋊4D4⋊36C2, D10.20(C4○D4), (C2×C20).194C23, (C2×C10).251C24, C4⋊Dic5.54C22, D10.12D4⋊51C2, D10.13D4⋊41C2, C2.64(D4⋊8D10), C23.57(C22×D5), (C2×D20).235C22, C22.D20⋊29C2, (C22×C10).65C23, C22.272(C23×D5), Dic5.14D4⋊45C2, C23.D5.67C22, C5⋊9(C22.33C24), (C2×Dic5).275C23, (C4×Dic5).159C22, C10.D4.56C22, (C22×D5).235C23, C2.64(D4.10D10), D10⋊C4.114C22, (C2×Dic10).190C22, (C22×Dic5).151C22, (D5×C4⋊C4)⋊41C2, C2.98(D5×C4○D4), C4⋊C4⋊D5⋊42C2, (C5×C42⋊2C2)⋊6C2, C10.209(C2×C4○D4), (C2×C4×D5).270C22, (C5×C4⋊C4).203C22, (C2×C4).209(C22×D5), (C2×C5⋊D4).71C22, (C5×C22⋊C4).76C22, SmallGroup(320,1379)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 798 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], D5 [×3], C10 [×3], C10, C42, C42, C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×11], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×6], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2, C42⋊2C2, Dic10, C4×D5 [×5], D20 [×2], C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×C10, C22.33C24, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×5], D10⋊C4 [×6], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], C20.6Q8, C4×D20, Dic5.14D4, Dic5⋊4D4, D10.12D4 [×2], D10⋊D4, C22.D20, C4.Dic10, D5×C4⋊C4, D10.13D4, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.161D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.33C24, C23×D5, D5×C4○D4, D4⋊8D10, D4.10D10, C42.161D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 34 70 159)(2 25 71 150)(3 36 72 141)(4 27 73 152)(5 38 74 143)(6 29 75 154)(7 40 76 145)(8 31 77 156)(9 22 78 147)(10 33 79 158)(11 24 80 149)(12 35 61 160)(13 26 62 151)(14 37 63 142)(15 28 64 153)(16 39 65 144)(17 30 66 155)(18 21 67 146)(19 32 68 157)(20 23 69 148)(41 81 107 121)(42 92 108 132)(43 83 109 123)(44 94 110 134)(45 85 111 125)(46 96 112 136)(47 87 113 127)(48 98 114 138)(49 89 115 129)(50 100 116 140)(51 91 117 131)(52 82 118 122)(53 93 119 133)(54 84 120 124)(55 95 101 135)(56 86 102 126)(57 97 103 137)(58 88 104 128)(59 99 105 139)(60 90 106 130)
(1 97 11 87)(2 128 12 138)(3 99 13 89)(4 130 14 140)(5 81 15 91)(6 132 16 122)(7 83 17 93)(8 134 18 124)(9 85 19 95)(10 136 20 126)(21 54 31 44)(22 111 32 101)(23 56 33 46)(24 113 34 103)(25 58 35 48)(26 115 36 105)(27 60 37 50)(28 117 38 107)(29 42 39 52)(30 119 40 109)(41 153 51 143)(43 155 53 145)(45 157 55 147)(47 159 57 149)(49 141 59 151)(61 98 71 88)(62 129 72 139)(63 100 73 90)(64 131 74 121)(65 82 75 92)(66 133 76 123)(67 84 77 94)(68 135 78 125)(69 86 79 96)(70 137 80 127)(102 158 112 148)(104 160 114 150)(106 142 116 152)(108 144 118 154)(110 146 120 156)
(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 96 11 86)(2 85 12 95)(3 94 13 84)(4 83 14 93)(5 92 15 82)(6 81 16 91)(7 90 17 100)(8 99 18 89)(9 88 19 98)(10 97 20 87)(21 105 31 115)(22 114 32 104)(23 103 33 113)(24 112 34 102)(25 101 35 111)(26 110 36 120)(27 119 37 109)(28 108 38 118)(29 117 39 107)(30 106 40 116)(41 154 51 144)(42 143 52 153)(43 152 53 142)(44 141 54 151)(45 150 55 160)(46 159 56 149)(47 148 57 158)(48 157 58 147)(49 146 59 156)(50 155 60 145)(61 135 71 125)(62 124 72 134)(63 133 73 123)(64 122 74 132)(65 131 75 121)(66 140 76 130)(67 129 77 139)(68 138 78 128)(69 127 79 137)(70 136 80 126)
G:=sub<Sym(160)| (1,34,70,159)(2,25,71,150)(3,36,72,141)(4,27,73,152)(5,38,74,143)(6,29,75,154)(7,40,76,145)(8,31,77,156)(9,22,78,147)(10,33,79,158)(11,24,80,149)(12,35,61,160)(13,26,62,151)(14,37,63,142)(15,28,64,153)(16,39,65,144)(17,30,66,155)(18,21,67,146)(19,32,68,157)(20,23,69,148)(41,81,107,121)(42,92,108,132)(43,83,109,123)(44,94,110,134)(45,85,111,125)(46,96,112,136)(47,87,113,127)(48,98,114,138)(49,89,115,129)(50,100,116,140)(51,91,117,131)(52,82,118,122)(53,93,119,133)(54,84,120,124)(55,95,101,135)(56,86,102,126)(57,97,103,137)(58,88,104,128)(59,99,105,139)(60,90,106,130), (1,97,11,87)(2,128,12,138)(3,99,13,89)(4,130,14,140)(5,81,15,91)(6,132,16,122)(7,83,17,93)(8,134,18,124)(9,85,19,95)(10,136,20,126)(21,54,31,44)(22,111,32,101)(23,56,33,46)(24,113,34,103)(25,58,35,48)(26,115,36,105)(27,60,37,50)(28,117,38,107)(29,42,39,52)(30,119,40,109)(41,153,51,143)(43,155,53,145)(45,157,55,147)(47,159,57,149)(49,141,59,151)(61,98,71,88)(62,129,72,139)(63,100,73,90)(64,131,74,121)(65,82,75,92)(66,133,76,123)(67,84,77,94)(68,135,78,125)(69,86,79,96)(70,137,80,127)(102,158,112,148)(104,160,114,150)(106,142,116,152)(108,144,118,154)(110,146,120,156), (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), (1,96,11,86)(2,85,12,95)(3,94,13,84)(4,83,14,93)(5,92,15,82)(6,81,16,91)(7,90,17,100)(8,99,18,89)(9,88,19,98)(10,97,20,87)(21,105,31,115)(22,114,32,104)(23,103,33,113)(24,112,34,102)(25,101,35,111)(26,110,36,120)(27,119,37,109)(28,108,38,118)(29,117,39,107)(30,106,40,116)(41,154,51,144)(42,143,52,153)(43,152,53,142)(44,141,54,151)(45,150,55,160)(46,159,56,149)(47,148,57,158)(48,157,58,147)(49,146,59,156)(50,155,60,145)(61,135,71,125)(62,124,72,134)(63,133,73,123)(64,122,74,132)(65,131,75,121)(66,140,76,130)(67,129,77,139)(68,138,78,128)(69,127,79,137)(70,136,80,126)>;
G:=Group( (1,34,70,159)(2,25,71,150)(3,36,72,141)(4,27,73,152)(5,38,74,143)(6,29,75,154)(7,40,76,145)(8,31,77,156)(9,22,78,147)(10,33,79,158)(11,24,80,149)(12,35,61,160)(13,26,62,151)(14,37,63,142)(15,28,64,153)(16,39,65,144)(17,30,66,155)(18,21,67,146)(19,32,68,157)(20,23,69,148)(41,81,107,121)(42,92,108,132)(43,83,109,123)(44,94,110,134)(45,85,111,125)(46,96,112,136)(47,87,113,127)(48,98,114,138)(49,89,115,129)(50,100,116,140)(51,91,117,131)(52,82,118,122)(53,93,119,133)(54,84,120,124)(55,95,101,135)(56,86,102,126)(57,97,103,137)(58,88,104,128)(59,99,105,139)(60,90,106,130), (1,97,11,87)(2,128,12,138)(3,99,13,89)(4,130,14,140)(5,81,15,91)(6,132,16,122)(7,83,17,93)(8,134,18,124)(9,85,19,95)(10,136,20,126)(21,54,31,44)(22,111,32,101)(23,56,33,46)(24,113,34,103)(25,58,35,48)(26,115,36,105)(27,60,37,50)(28,117,38,107)(29,42,39,52)(30,119,40,109)(41,153,51,143)(43,155,53,145)(45,157,55,147)(47,159,57,149)(49,141,59,151)(61,98,71,88)(62,129,72,139)(63,100,73,90)(64,131,74,121)(65,82,75,92)(66,133,76,123)(67,84,77,94)(68,135,78,125)(69,86,79,96)(70,137,80,127)(102,158,112,148)(104,160,114,150)(106,142,116,152)(108,144,118,154)(110,146,120,156), (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), (1,96,11,86)(2,85,12,95)(3,94,13,84)(4,83,14,93)(5,92,15,82)(6,81,16,91)(7,90,17,100)(8,99,18,89)(9,88,19,98)(10,97,20,87)(21,105,31,115)(22,114,32,104)(23,103,33,113)(24,112,34,102)(25,101,35,111)(26,110,36,120)(27,119,37,109)(28,108,38,118)(29,117,39,107)(30,106,40,116)(41,154,51,144)(42,143,52,153)(43,152,53,142)(44,141,54,151)(45,150,55,160)(46,159,56,149)(47,148,57,158)(48,157,58,147)(49,146,59,156)(50,155,60,145)(61,135,71,125)(62,124,72,134)(63,133,73,123)(64,122,74,132)(65,131,75,121)(66,140,76,130)(67,129,77,139)(68,138,78,128)(69,127,79,137)(70,136,80,126) );
G=PermutationGroup([(1,34,70,159),(2,25,71,150),(3,36,72,141),(4,27,73,152),(5,38,74,143),(6,29,75,154),(7,40,76,145),(8,31,77,156),(9,22,78,147),(10,33,79,158),(11,24,80,149),(12,35,61,160),(13,26,62,151),(14,37,63,142),(15,28,64,153),(16,39,65,144),(17,30,66,155),(18,21,67,146),(19,32,68,157),(20,23,69,148),(41,81,107,121),(42,92,108,132),(43,83,109,123),(44,94,110,134),(45,85,111,125),(46,96,112,136),(47,87,113,127),(48,98,114,138),(49,89,115,129),(50,100,116,140),(51,91,117,131),(52,82,118,122),(53,93,119,133),(54,84,120,124),(55,95,101,135),(56,86,102,126),(57,97,103,137),(58,88,104,128),(59,99,105,139),(60,90,106,130)], [(1,97,11,87),(2,128,12,138),(3,99,13,89),(4,130,14,140),(5,81,15,91),(6,132,16,122),(7,83,17,93),(8,134,18,124),(9,85,19,95),(10,136,20,126),(21,54,31,44),(22,111,32,101),(23,56,33,46),(24,113,34,103),(25,58,35,48),(26,115,36,105),(27,60,37,50),(28,117,38,107),(29,42,39,52),(30,119,40,109),(41,153,51,143),(43,155,53,145),(45,157,55,147),(47,159,57,149),(49,141,59,151),(61,98,71,88),(62,129,72,139),(63,100,73,90),(64,131,74,121),(65,82,75,92),(66,133,76,123),(67,84,77,94),(68,135,78,125),(69,86,79,96),(70,137,80,127),(102,158,112,148),(104,160,114,150),(106,142,116,152),(108,144,118,154),(110,146,120,156)], [(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)], [(1,96,11,86),(2,85,12,95),(3,94,13,84),(4,83,14,93),(5,92,15,82),(6,81,16,91),(7,90,17,100),(8,99,18,89),(9,88,19,98),(10,97,20,87),(21,105,31,115),(22,114,32,104),(23,103,33,113),(24,112,34,102),(25,101,35,111),(26,110,36,120),(27,119,37,109),(28,108,38,118),(29,117,39,107),(30,106,40,116),(41,154,51,144),(42,143,52,153),(43,152,53,142),(44,141,54,151),(45,150,55,160),(46,159,56,149),(47,148,57,158),(48,157,58,147),(49,146,59,156),(50,155,60,145),(61,135,71,125),(62,124,72,134),(63,133,73,123),(64,122,74,132),(65,131,75,121),(66,140,76,130),(67,129,77,139),(68,138,78,128),(69,127,79,137),(70,136,80,126)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 39 | 0 |
| 0 | 0 | 0 | 1 | 0 | 39 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 33 | 2 | 0 | 0 | 0 | 0 |
| 30 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 13 | 13 |
| 0 | 0 | 26 | 33 | 28 | 4 |
| 0 | 0 | 1 | 1 | 26 | 26 |
| 0 | 0 | 40 | 35 | 15 | 8 |
| 33 | 2 | 0 | 0 | 0 | 0 |
| 30 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 30 | 15 | 26 |
| 0 | 0 | 35 | 30 | 3 | 26 |
| 0 | 0 | 39 | 2 | 30 | 11 |
| 0 | 0 | 16 | 2 | 6 | 11 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[33,30,0,0,0,0,2,8,0,0,0,0,0,0,15,26,1,40,0,0,15,33,1,35,0,0,13,28,26,15,0,0,13,4,26,8],[33,30,0,0,0,0,2,8,0,0,0,0,0,0,11,35,39,16,0,0,30,30,2,2,0,0,15,3,30,6,0,0,26,26,11,11] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D5×C4○D4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.161D10 | C20.6Q8 | C4×D20 | Dic5.14D4 | Dic5⋊4D4 | D10.12D4 | D10⋊D4 | C22.D20 | C4.Dic10 | D5×C4⋊C4 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 6 | 6 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{161}D_{10} % in TeX
G:=Group("C4^2.161D10"); // GroupNames label
G:=SmallGroup(320,1379);
// by ID
G=gap.SmallGroup(320,1379);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀