metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.158D10, C10.322- (1+4), C10.1362+ (1+4), C4⋊2D20⋊35C2, C4⋊C4.115D10, C42.C2⋊14D5, D10⋊Q8⋊38C2, C4.D20⋊32C2, (C4×C20).225C22, (C2×C10).244C24, (C2×C20).191C23, D10.13D4⋊37C2, C2.61(D4⋊8D10), (C2×D20).173C22, C22.265(C23×D5), D10⋊C4.74C22, C5⋊5(C22.56C24), (C2×Dic10).44C22, (C2×Dic5).126C23, C10.D4.55C22, (C22×D5).109C23, C2.33(Q8.10D10), (C5×C42.C2)⋊17C2, (C2×C4×D5).143C22, (C5×C4⋊C4).199C22, (C2×C4).208(C22×D5), SmallGroup(320,1372)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 950 in 220 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23 [×4], D5 [×4], C10, C10 [×2], C42, C22⋊C4 [×12], C4⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic5 [×4], C20 [×7], D10 [×12], C2×C10, C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic10 [×2], C4×D5 [×4], D20 [×6], C2×Dic5 [×4], C2×C20, C2×C20 [×6], C22×D5 [×4], C22.56C24, C10.D4 [×4], D10⋊C4 [×12], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×6], C4.D20 [×2], D10.13D4 [×4], C4⋊2D20 [×4], D10⋊Q8 [×4], C5×C42.C2, C42.158D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4) [×2], 2- (1+4), C22×D5 [×7], C22.56C24, C23×D5, Q8.10D10, D4⋊8D10 [×2], C42.158D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 93 11 83)(2 153 12 143)(3 95 13 85)(4 155 14 145)(5 97 15 87)(6 157 16 147)(7 99 17 89)(8 159 18 149)(9 81 19 91)(10 141 20 151)(21 142 31 152)(22 84 32 94)(23 144 33 154)(24 86 34 96)(25 146 35 156)(26 88 36 98)(27 148 37 158)(28 90 38 100)(29 150 39 160)(30 92 40 82)(41 129 51 139)(42 73 52 63)(43 131 53 121)(44 75 54 65)(45 133 55 123)(46 77 56 67)(47 135 57 125)(48 79 58 69)(49 137 59 127)(50 61 60 71)(62 109 72 119)(64 111 74 101)(66 113 76 103)(68 115 78 105)(70 117 80 107)(102 122 112 132)(104 124 114 134)(106 126 116 136)(108 128 118 138)(110 130 120 140)
(1 140 21 73)(2 74 22 121)(3 122 23 75)(4 76 24 123)(5 124 25 77)(6 78 26 125)(7 126 27 79)(8 80 28 127)(9 128 29 61)(10 62 30 129)(11 130 31 63)(12 64 32 131)(13 132 33 65)(14 66 34 133)(15 134 35 67)(16 68 36 135)(17 136 37 69)(18 70 38 137)(19 138 39 71)(20 72 40 139)(41 151 119 82)(42 83 120 152)(43 153 101 84)(44 85 102 154)(45 155 103 86)(46 87 104 156)(47 157 105 88)(48 89 106 158)(49 159 107 90)(50 91 108 160)(51 141 109 92)(52 93 110 142)(53 143 111 94)(54 95 112 144)(55 145 113 96)(56 97 114 146)(57 147 115 98)(58 99 116 148)(59 149 117 100)(60 81 118 150)
(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 77 11 67)(2 66 12 76)(3 75 13 65)(4 64 14 74)(5 73 15 63)(6 62 16 72)(7 71 17 61)(8 80 18 70)(9 69 19 79)(10 78 20 68)(21 124 31 134)(22 133 32 123)(23 122 33 132)(24 131 34 121)(25 140 35 130)(26 129 36 139)(27 138 37 128)(28 127 38 137)(29 136 39 126)(30 125 40 135)(41 98 51 88)(42 87 52 97)(43 96 53 86)(44 85 54 95)(45 94 55 84)(46 83 56 93)(47 92 57 82)(48 81 58 91)(49 90 59 100)(50 99 60 89)(101 145 111 155)(102 154 112 144)(103 143 113 153)(104 152 114 142)(105 141 115 151)(106 150 116 160)(107 159 117 149)(108 148 118 158)(109 157 119 147)(110 146 120 156)
G:=sub<Sym(160)| (1,93,11,83)(2,153,12,143)(3,95,13,85)(4,155,14,145)(5,97,15,87)(6,157,16,147)(7,99,17,89)(8,159,18,149)(9,81,19,91)(10,141,20,151)(21,142,31,152)(22,84,32,94)(23,144,33,154)(24,86,34,96)(25,146,35,156)(26,88,36,98)(27,148,37,158)(28,90,38,100)(29,150,39,160)(30,92,40,82)(41,129,51,139)(42,73,52,63)(43,131,53,121)(44,75,54,65)(45,133,55,123)(46,77,56,67)(47,135,57,125)(48,79,58,69)(49,137,59,127)(50,61,60,71)(62,109,72,119)(64,111,74,101)(66,113,76,103)(68,115,78,105)(70,117,80,107)(102,122,112,132)(104,124,114,134)(106,126,116,136)(108,128,118,138)(110,130,120,140), (1,140,21,73)(2,74,22,121)(3,122,23,75)(4,76,24,123)(5,124,25,77)(6,78,26,125)(7,126,27,79)(8,80,28,127)(9,128,29,61)(10,62,30,129)(11,130,31,63)(12,64,32,131)(13,132,33,65)(14,66,34,133)(15,134,35,67)(16,68,36,135)(17,136,37,69)(18,70,38,137)(19,138,39,71)(20,72,40,139)(41,151,119,82)(42,83,120,152)(43,153,101,84)(44,85,102,154)(45,155,103,86)(46,87,104,156)(47,157,105,88)(48,89,106,158)(49,159,107,90)(50,91,108,160)(51,141,109,92)(52,93,110,142)(53,143,111,94)(54,95,112,144)(55,145,113,96)(56,97,114,146)(57,147,115,98)(58,99,116,148)(59,149,117,100)(60,81,118,150), (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), (1,77,11,67)(2,66,12,76)(3,75,13,65)(4,64,14,74)(5,73,15,63)(6,62,16,72)(7,71,17,61)(8,80,18,70)(9,69,19,79)(10,78,20,68)(21,124,31,134)(22,133,32,123)(23,122,33,132)(24,131,34,121)(25,140,35,130)(26,129,36,139)(27,138,37,128)(28,127,38,137)(29,136,39,126)(30,125,40,135)(41,98,51,88)(42,87,52,97)(43,96,53,86)(44,85,54,95)(45,94,55,84)(46,83,56,93)(47,92,57,82)(48,81,58,91)(49,90,59,100)(50,99,60,89)(101,145,111,155)(102,154,112,144)(103,143,113,153)(104,152,114,142)(105,141,115,151)(106,150,116,160)(107,159,117,149)(108,148,118,158)(109,157,119,147)(110,146,120,156)>;
G:=Group( (1,93,11,83)(2,153,12,143)(3,95,13,85)(4,155,14,145)(5,97,15,87)(6,157,16,147)(7,99,17,89)(8,159,18,149)(9,81,19,91)(10,141,20,151)(21,142,31,152)(22,84,32,94)(23,144,33,154)(24,86,34,96)(25,146,35,156)(26,88,36,98)(27,148,37,158)(28,90,38,100)(29,150,39,160)(30,92,40,82)(41,129,51,139)(42,73,52,63)(43,131,53,121)(44,75,54,65)(45,133,55,123)(46,77,56,67)(47,135,57,125)(48,79,58,69)(49,137,59,127)(50,61,60,71)(62,109,72,119)(64,111,74,101)(66,113,76,103)(68,115,78,105)(70,117,80,107)(102,122,112,132)(104,124,114,134)(106,126,116,136)(108,128,118,138)(110,130,120,140), (1,140,21,73)(2,74,22,121)(3,122,23,75)(4,76,24,123)(5,124,25,77)(6,78,26,125)(7,126,27,79)(8,80,28,127)(9,128,29,61)(10,62,30,129)(11,130,31,63)(12,64,32,131)(13,132,33,65)(14,66,34,133)(15,134,35,67)(16,68,36,135)(17,136,37,69)(18,70,38,137)(19,138,39,71)(20,72,40,139)(41,151,119,82)(42,83,120,152)(43,153,101,84)(44,85,102,154)(45,155,103,86)(46,87,104,156)(47,157,105,88)(48,89,106,158)(49,159,107,90)(50,91,108,160)(51,141,109,92)(52,93,110,142)(53,143,111,94)(54,95,112,144)(55,145,113,96)(56,97,114,146)(57,147,115,98)(58,99,116,148)(59,149,117,100)(60,81,118,150), (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), (1,77,11,67)(2,66,12,76)(3,75,13,65)(4,64,14,74)(5,73,15,63)(6,62,16,72)(7,71,17,61)(8,80,18,70)(9,69,19,79)(10,78,20,68)(21,124,31,134)(22,133,32,123)(23,122,33,132)(24,131,34,121)(25,140,35,130)(26,129,36,139)(27,138,37,128)(28,127,38,137)(29,136,39,126)(30,125,40,135)(41,98,51,88)(42,87,52,97)(43,96,53,86)(44,85,54,95)(45,94,55,84)(46,83,56,93)(47,92,57,82)(48,81,58,91)(49,90,59,100)(50,99,60,89)(101,145,111,155)(102,154,112,144)(103,143,113,153)(104,152,114,142)(105,141,115,151)(106,150,116,160)(107,159,117,149)(108,148,118,158)(109,157,119,147)(110,146,120,156) );
G=PermutationGroup([(1,93,11,83),(2,153,12,143),(3,95,13,85),(4,155,14,145),(5,97,15,87),(6,157,16,147),(7,99,17,89),(8,159,18,149),(9,81,19,91),(10,141,20,151),(21,142,31,152),(22,84,32,94),(23,144,33,154),(24,86,34,96),(25,146,35,156),(26,88,36,98),(27,148,37,158),(28,90,38,100),(29,150,39,160),(30,92,40,82),(41,129,51,139),(42,73,52,63),(43,131,53,121),(44,75,54,65),(45,133,55,123),(46,77,56,67),(47,135,57,125),(48,79,58,69),(49,137,59,127),(50,61,60,71),(62,109,72,119),(64,111,74,101),(66,113,76,103),(68,115,78,105),(70,117,80,107),(102,122,112,132),(104,124,114,134),(106,126,116,136),(108,128,118,138),(110,130,120,140)], [(1,140,21,73),(2,74,22,121),(3,122,23,75),(4,76,24,123),(5,124,25,77),(6,78,26,125),(7,126,27,79),(8,80,28,127),(9,128,29,61),(10,62,30,129),(11,130,31,63),(12,64,32,131),(13,132,33,65),(14,66,34,133),(15,134,35,67),(16,68,36,135),(17,136,37,69),(18,70,38,137),(19,138,39,71),(20,72,40,139),(41,151,119,82),(42,83,120,152),(43,153,101,84),(44,85,102,154),(45,155,103,86),(46,87,104,156),(47,157,105,88),(48,89,106,158),(49,159,107,90),(50,91,108,160),(51,141,109,92),(52,93,110,142),(53,143,111,94),(54,95,112,144),(55,145,113,96),(56,97,114,146),(57,147,115,98),(58,99,116,148),(59,149,117,100),(60,81,118,150)], [(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)], [(1,77,11,67),(2,66,12,76),(3,75,13,65),(4,64,14,74),(5,73,15,63),(6,62,16,72),(7,71,17,61),(8,80,18,70),(9,69,19,79),(10,78,20,68),(21,124,31,134),(22,133,32,123),(23,122,33,132),(24,131,34,121),(25,140,35,130),(26,129,36,139),(27,138,37,128),(28,127,38,137),(29,136,39,126),(30,125,40,135),(41,98,51,88),(42,87,52,97),(43,96,53,86),(44,85,54,95),(45,94,55,84),(46,83,56,93),(47,92,57,82),(48,81,58,91),(49,90,59,100),(50,99,60,89),(101,145,111,155),(102,154,112,144),(103,143,113,153),(104,152,114,142),(105,141,115,151),(106,150,116,160),(107,159,117,149),(108,148,118,158),(109,157,119,147),(110,146,120,156)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 11 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 13 | 20 | 26 |
| 0 | 0 | 0 | 0 | 28 | 2 | 21 | 5 |
| 0 | 0 | 0 | 0 | 2 | 6 | 11 | 28 |
| 0 | 0 | 0 | 0 | 8 | 8 | 22 | 30 |
| 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 | 40 | 0 | 28 | 13 |
| 0 | 0 | 0 | 0 | 0 | 40 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 38 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 38 | 0 | 1 |
| 20 | 21 | 40 | 1 | 0 | 0 | 0 | 0 |
| 20 | 37 | 40 | 33 | 0 | 0 | 0 | 0 |
| 40 | 1 | 21 | 20 | 0 | 0 | 0 | 0 |
| 40 | 33 | 21 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 28 | 34 | 1 |
| 0 | 0 | 0 | 0 | 13 | 31 | 0 | 35 |
| 0 | 0 | 0 | 0 | 25 | 25 | 38 | 13 |
| 0 | 0 | 0 | 0 | 27 | 9 | 19 | 19 |
| 0 | 0 | 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 13 | 28 |
| 0 | 0 | 0 | 0 | 35 | 40 | 32 | 37 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 34 |
G:=sub<GL(8,GF(41))| [11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,39,28,2,8,0,0,0,0,13,2,6,8,0,0,0,0,20,21,11,22,0,0,0,0,26,5,28,30],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,3,0,0,0,0,0,40,38,38,0,0,0,0,28,28,1,0,0,0,0,0,13,0,0,1],[20,20,40,40,0,0,0,0,21,37,1,33,0,0,0,0,40,40,21,21,0,0,0,0,1,33,20,4,0,0,0,0,0,0,0,0,35,13,25,27,0,0,0,0,28,31,25,9,0,0,0,0,34,0,38,19,0,0,0,0,1,35,13,19],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,13,32,7,8,0,0,0,0,28,37,35,34] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2+ (1+4) | 2- (1+4) | Q8.10D10 | D4⋊8D10 |
| kernel | C42.158D10 | C4.D20 | D10.13D4 | C4⋊2D20 | D10⋊Q8 | C5×C42.C2 | C42.C2 | C42 | C4⋊C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 12 | 2 | 1 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{158}D_{10} % in TeX
G:=Group("C4^2.158D10"); // GroupNames label
G:=SmallGroup(320,1372);
// by ID
G=gap.SmallGroup(320,1372);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,675,570,136,12550]);
// 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^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀