metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.159D10, C10.982- (1+4), C20⋊Q8⋊39C2, C42⋊2C2.D5, C4⋊C4.116D10, C20.6Q8⋊8C2, (C4×Dic10)⋊13C2, (C2×C20).93C23, (C4×C20).31C22, C22⋊C4.39D10, C4.Dic10⋊38C2, Dic5⋊3Q8⋊39C2, (C2×C10).245C24, C4⋊Dic5.53C22, C23.51(C22×D5), Dic5.20(C4○D4), Dic5.Q8⋊36C2, (C22×C10).59C23, C23.D10.3C2, C22.266(C23×D5), C23.D5.61C22, Dic5.14D4.4C2, C5⋊6(C22.35C24), (C2×Dic5).127C23, (C4×Dic5).237C22, C23.11D10.3C2, C2.62(D4.10D10), (C2×Dic10).261C22, C10.D4.126C22, (C22×Dic5).148C22, C2.92(D5×C4○D4), C10.203(C2×C4○D4), (C5×C4⋊C4).200C22, (C5×C42⋊2C2).1C2, (C2×C4).302(C22×D5), (C5×C22⋊C4).70C22, SmallGroup(320,1373)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 558 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2, C4 [×15], C22, C22 [×3], C5, C2×C4 [×6], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10, C42, C42 [×5], C22⋊C4 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×17], C22×C4, C2×Q8 [×2], Dic5 [×2], Dic5 [×7], C20 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×6], C22×C10, C22.35C24, C4×Dic5 [×5], C10.D4 [×12], C4⋊Dic5 [×5], C23.D5 [×3], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×2], C22×Dic5, C4×Dic10, C20.6Q8, C23.11D10, Dic5.14D4 [×2], C23.D10 [×3], Dic5⋊3Q8, C20⋊Q8, Dic5.Q8 [×3], C4.Dic10, C5×C42⋊2C2, C42.159D10
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], C22.35C24, C23×D5, D5×C4○D4, D4.10D10 [×2], C42.159D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 82 127 53)(2 93 128 44)(3 84 129 55)(4 95 130 46)(5 86 131 57)(6 97 132 48)(7 88 133 59)(8 99 134 50)(9 90 135 41)(10 81 136 52)(11 92 137 43)(12 83 138 54)(13 94 139 45)(14 85 140 56)(15 96 121 47)(16 87 122 58)(17 98 123 49)(18 89 124 60)(19 100 125 51)(20 91 126 42)(21 107 68 146)(22 118 69 157)(23 109 70 148)(24 120 71 159)(25 111 72 150)(26 102 73 141)(27 113 74 152)(28 104 75 143)(29 115 76 154)(30 106 77 145)(31 117 78 156)(32 108 79 147)(33 119 80 158)(34 110 61 149)(35 101 62 160)(36 112 63 151)(37 103 64 142)(38 114 65 153)(39 105 66 144)(40 116 67 155)
(1 87 11 97)(2 49 12 59)(3 89 13 99)(4 51 14 41)(5 91 15 81)(6 53 16 43)(7 93 17 83)(8 55 18 45)(9 95 19 85)(10 57 20 47)(21 151 31 141)(22 103 32 113)(23 153 33 143)(24 105 34 115)(25 155 35 145)(26 107 36 117)(27 157 37 147)(28 109 38 119)(29 159 39 149)(30 111 40 101)(42 121 52 131)(44 123 54 133)(46 125 56 135)(48 127 58 137)(50 129 60 139)(61 154 71 144)(62 106 72 116)(63 156 73 146)(64 108 74 118)(65 158 75 148)(66 110 76 120)(67 160 77 150)(68 112 78 102)(69 142 79 152)(70 114 80 104)(82 122 92 132)(84 124 94 134)(86 126 96 136)(88 128 98 138)(90 130 100 140)
(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)
(1 152 127 113)(2 141 128 102)(3 150 129 111)(4 159 130 120)(5 148 131 109)(6 157 132 118)(7 146 133 107)(8 155 134 116)(9 144 135 105)(10 153 136 114)(11 142 137 103)(12 151 138 112)(13 160 139 101)(14 149 140 110)(15 158 121 119)(16 147 122 108)(17 156 123 117)(18 145 124 106)(19 154 125 115)(20 143 126 104)(21 49 68 98)(22 58 69 87)(23 47 70 96)(24 56 71 85)(25 45 72 94)(26 54 73 83)(27 43 74 92)(28 52 75 81)(29 41 76 90)(30 50 77 99)(31 59 78 88)(32 48 79 97)(33 57 80 86)(34 46 61 95)(35 55 62 84)(36 44 63 93)(37 53 64 82)(38 42 65 91)(39 51 66 100)(40 60 67 89)
G:=sub<Sym(160)| (1,82,127,53)(2,93,128,44)(3,84,129,55)(4,95,130,46)(5,86,131,57)(6,97,132,48)(7,88,133,59)(8,99,134,50)(9,90,135,41)(10,81,136,52)(11,92,137,43)(12,83,138,54)(13,94,139,45)(14,85,140,56)(15,96,121,47)(16,87,122,58)(17,98,123,49)(18,89,124,60)(19,100,125,51)(20,91,126,42)(21,107,68,146)(22,118,69,157)(23,109,70,148)(24,120,71,159)(25,111,72,150)(26,102,73,141)(27,113,74,152)(28,104,75,143)(29,115,76,154)(30,106,77,145)(31,117,78,156)(32,108,79,147)(33,119,80,158)(34,110,61,149)(35,101,62,160)(36,112,63,151)(37,103,64,142)(38,114,65,153)(39,105,66,144)(40,116,67,155), (1,87,11,97)(2,49,12,59)(3,89,13,99)(4,51,14,41)(5,91,15,81)(6,53,16,43)(7,93,17,83)(8,55,18,45)(9,95,19,85)(10,57,20,47)(21,151,31,141)(22,103,32,113)(23,153,33,143)(24,105,34,115)(25,155,35,145)(26,107,36,117)(27,157,37,147)(28,109,38,119)(29,159,39,149)(30,111,40,101)(42,121,52,131)(44,123,54,133)(46,125,56,135)(48,127,58,137)(50,129,60,139)(61,154,71,144)(62,106,72,116)(63,156,73,146)(64,108,74,118)(65,158,75,148)(66,110,76,120)(67,160,77,150)(68,112,78,102)(69,142,79,152)(70,114,80,104)(82,122,92,132)(84,124,94,134)(86,126,96,136)(88,128,98,138)(90,130,100,140), (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), (1,152,127,113)(2,141,128,102)(3,150,129,111)(4,159,130,120)(5,148,131,109)(6,157,132,118)(7,146,133,107)(8,155,134,116)(9,144,135,105)(10,153,136,114)(11,142,137,103)(12,151,138,112)(13,160,139,101)(14,149,140,110)(15,158,121,119)(16,147,122,108)(17,156,123,117)(18,145,124,106)(19,154,125,115)(20,143,126,104)(21,49,68,98)(22,58,69,87)(23,47,70,96)(24,56,71,85)(25,45,72,94)(26,54,73,83)(27,43,74,92)(28,52,75,81)(29,41,76,90)(30,50,77,99)(31,59,78,88)(32,48,79,97)(33,57,80,86)(34,46,61,95)(35,55,62,84)(36,44,63,93)(37,53,64,82)(38,42,65,91)(39,51,66,100)(40,60,67,89)>;
G:=Group( (1,82,127,53)(2,93,128,44)(3,84,129,55)(4,95,130,46)(5,86,131,57)(6,97,132,48)(7,88,133,59)(8,99,134,50)(9,90,135,41)(10,81,136,52)(11,92,137,43)(12,83,138,54)(13,94,139,45)(14,85,140,56)(15,96,121,47)(16,87,122,58)(17,98,123,49)(18,89,124,60)(19,100,125,51)(20,91,126,42)(21,107,68,146)(22,118,69,157)(23,109,70,148)(24,120,71,159)(25,111,72,150)(26,102,73,141)(27,113,74,152)(28,104,75,143)(29,115,76,154)(30,106,77,145)(31,117,78,156)(32,108,79,147)(33,119,80,158)(34,110,61,149)(35,101,62,160)(36,112,63,151)(37,103,64,142)(38,114,65,153)(39,105,66,144)(40,116,67,155), (1,87,11,97)(2,49,12,59)(3,89,13,99)(4,51,14,41)(5,91,15,81)(6,53,16,43)(7,93,17,83)(8,55,18,45)(9,95,19,85)(10,57,20,47)(21,151,31,141)(22,103,32,113)(23,153,33,143)(24,105,34,115)(25,155,35,145)(26,107,36,117)(27,157,37,147)(28,109,38,119)(29,159,39,149)(30,111,40,101)(42,121,52,131)(44,123,54,133)(46,125,56,135)(48,127,58,137)(50,129,60,139)(61,154,71,144)(62,106,72,116)(63,156,73,146)(64,108,74,118)(65,158,75,148)(66,110,76,120)(67,160,77,150)(68,112,78,102)(69,142,79,152)(70,114,80,104)(82,122,92,132)(84,124,94,134)(86,126,96,136)(88,128,98,138)(90,130,100,140), (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), (1,152,127,113)(2,141,128,102)(3,150,129,111)(4,159,130,120)(5,148,131,109)(6,157,132,118)(7,146,133,107)(8,155,134,116)(9,144,135,105)(10,153,136,114)(11,142,137,103)(12,151,138,112)(13,160,139,101)(14,149,140,110)(15,158,121,119)(16,147,122,108)(17,156,123,117)(18,145,124,106)(19,154,125,115)(20,143,126,104)(21,49,68,98)(22,58,69,87)(23,47,70,96)(24,56,71,85)(25,45,72,94)(26,54,73,83)(27,43,74,92)(28,52,75,81)(29,41,76,90)(30,50,77,99)(31,59,78,88)(32,48,79,97)(33,57,80,86)(34,46,61,95)(35,55,62,84)(36,44,63,93)(37,53,64,82)(38,42,65,91)(39,51,66,100)(40,60,67,89) );
G=PermutationGroup([(1,82,127,53),(2,93,128,44),(3,84,129,55),(4,95,130,46),(5,86,131,57),(6,97,132,48),(7,88,133,59),(8,99,134,50),(9,90,135,41),(10,81,136,52),(11,92,137,43),(12,83,138,54),(13,94,139,45),(14,85,140,56),(15,96,121,47),(16,87,122,58),(17,98,123,49),(18,89,124,60),(19,100,125,51),(20,91,126,42),(21,107,68,146),(22,118,69,157),(23,109,70,148),(24,120,71,159),(25,111,72,150),(26,102,73,141),(27,113,74,152),(28,104,75,143),(29,115,76,154),(30,106,77,145),(31,117,78,156),(32,108,79,147),(33,119,80,158),(34,110,61,149),(35,101,62,160),(36,112,63,151),(37,103,64,142),(38,114,65,153),(39,105,66,144),(40,116,67,155)], [(1,87,11,97),(2,49,12,59),(3,89,13,99),(4,51,14,41),(5,91,15,81),(6,53,16,43),(7,93,17,83),(8,55,18,45),(9,95,19,85),(10,57,20,47),(21,151,31,141),(22,103,32,113),(23,153,33,143),(24,105,34,115),(25,155,35,145),(26,107,36,117),(27,157,37,147),(28,109,38,119),(29,159,39,149),(30,111,40,101),(42,121,52,131),(44,123,54,133),(46,125,56,135),(48,127,58,137),(50,129,60,139),(61,154,71,144),(62,106,72,116),(63,156,73,146),(64,108,74,118),(65,158,75,148),(66,110,76,120),(67,160,77,150),(68,112,78,102),(69,142,79,152),(70,114,80,104),(82,122,92,132),(84,124,94,134),(86,126,96,136),(88,128,98,138),(90,130,100,140)], [(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)], [(1,152,127,113),(2,141,128,102),(3,150,129,111),(4,159,130,120),(5,148,131,109),(6,157,132,118),(7,146,133,107),(8,155,134,116),(9,144,135,105),(10,153,136,114),(11,142,137,103),(12,151,138,112),(13,160,139,101),(14,149,140,110),(15,158,121,119),(16,147,122,108),(17,156,123,117),(18,145,124,106),(19,154,125,115),(20,143,126,104),(21,49,68,98),(22,58,69,87),(23,47,70,96),(24,56,71,85),(25,45,72,94),(26,54,73,83),(27,43,74,92),(28,52,75,81),(29,41,76,90),(30,50,77,99),(31,59,78,88),(32,48,79,97),(33,57,80,86),(34,46,61,95),(35,55,62,84),(36,44,63,93),(37,53,64,82),(38,42,65,91),(39,51,66,100),(40,60,67,89)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 14 |
| 0 | 0 | 0 | 1 | 11 | 11 |
| 0 | 0 | 28 | 24 | 40 | 0 |
| 0 | 0 | 13 | 28 | 0 | 40 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 13 |
| 0 | 0 | 0 | 0 | 28 | 2 |
| 32 | 23 | 0 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 25 | 0 | 7 |
| 0 | 0 | 16 | 2 | 34 | 0 |
| 0 | 0 | 0 | 0 | 39 | 16 |
| 0 | 0 | 0 | 0 | 25 | 25 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 14 | 0 | 0 |
| 0 | 0 | 26 | 39 | 0 | 0 |
| 0 | 0 | 24 | 16 | 15 | 10 |
| 0 | 0 | 16 | 3 | 2 | 26 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,28,13,0,0,0,1,24,28,0,0,11,11,40,0,0,0,14,11,0,40],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,39,28,0,0,0,0,13,2],[32,9,0,0,0,0,23,9,0,0,0,0,0,0,16,16,0,0,0,0,25,2,0,0,0,0,0,34,39,25,0,0,7,0,16,25],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,2,26,24,16,0,0,14,39,16,3,0,0,0,0,15,2,0,0,0,0,10,26] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 4 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D5×C4○D4 | D4.10D10 |
| kernel | C42.159D10 | C4×Dic10 | C20.6Q8 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 1 | 2 | 4 | 2 | 6 | 6 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{159}D_{10} % in TeX
G:=Group("C4^2.159D10"); // GroupNames label
G:=SmallGroup(320,1373);
// by ID
G=gap.SmallGroup(320,1373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^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
○
≀