metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.89D10, C10.472- (1+4), C20⋊2Q8⋊5C2, C4⋊C4.267D10, (C4×Dic10)⋊7C2, C20.6Q8⋊3C2, (C2×C10).62C24, (C4×C20).22C22, C22⋊C4.90D10, C4.119(C4○D20), C20.235(C4○D4), (C2×C20).141C23, Dic5.Q8⋊4C2, C42⋊C2.12D5, (C22×C4).186D10, C4⋊Dic5.31C22, C22.95(C23×D5), C23.83(C22×D5), C20.48D4.18C2, C23.D5.3C22, (C2×Dic5).21C23, C23.D10.1C2, C2.6(D4.10D10), (C22×C10).132C23, (C22×C20).307C22, C5⋊1(C22.35C24), (C4×Dic5).214C22, C10.D4.74C22, (C2×Dic10).237C22, C2.29(C2×C4○D20), C10.27(C2×C4○D4), (C5×C4⋊C4).303C22, (C2×C4).269(C22×D5), (C5×C42⋊C2).13C2, (C5×C22⋊C4).111C22, SmallGroup(320,1190)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 542 in 192 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, C10, C10 [×2], C10, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C2×Q8 [×2], Dic5 [×8], C20 [×2], C20 [×5], C2×C10, C2×C10 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×4], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×C10, C22.35C24, C4×Dic5 [×4], C10.D4 [×12], C4⋊Dic5 [×6], C23.D5 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C22×C20, C4×Dic10 [×2], C20⋊2Q8, C20.6Q8, C23.D10 [×4], Dic5.Q8 [×4], C20.48D4 [×2], C5×C42⋊C2, C42.89D10
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, C4○D20 [×2], C23×D5, C2×C4○D20, D4.10D10 [×2], C42.89D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 85 31 95)(22 86 32 96)(23 87 33 97)(24 88 34 98)(25 89 35 99)(26 90 36 100)(27 91 37 81)(28 92 38 82)(29 93 39 83)(30 94 40 84)(41 159 51 149)(42 160 52 150)(43 141 53 151)(44 142 54 152)(45 143 55 153)(46 144 56 154)(47 145 57 155)(48 146 58 156)(49 147 59 157)(50 148 60 158)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)
(1 118 63 140)(2 109 64 131)(3 120 65 122)(4 111 66 133)(5 102 67 124)(6 113 68 135)(7 104 69 126)(8 115 70 137)(9 106 71 128)(10 117 72 139)(11 108 73 130)(12 119 74 121)(13 110 75 132)(14 101 76 123)(15 112 77 134)(16 103 78 125)(17 114 79 136)(18 105 80 127)(19 116 61 138)(20 107 62 129)(21 48 100 151)(22 59 81 142)(23 50 82 153)(24 41 83 144)(25 52 84 155)(26 43 85 146)(27 54 86 157)(28 45 87 148)(29 56 88 159)(30 47 89 150)(31 58 90 141)(32 49 91 152)(33 60 92 143)(34 51 93 154)(35 42 94 145)(36 53 95 156)(37 44 96 147)(38 55 97 158)(39 46 98 149)(40 57 99 160)
(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 38 63 97)(2 86 64 27)(3 36 65 95)(4 84 66 25)(5 34 67 93)(6 82 68 23)(7 32 69 91)(8 100 70 21)(9 30 71 89)(10 98 72 39)(11 28 73 87)(12 96 74 37)(13 26 75 85)(14 94 76 35)(15 24 77 83)(16 92 78 33)(17 22 79 81)(18 90 80 31)(19 40 61 99)(20 88 62 29)(41 124 144 102)(42 111 145 133)(43 122 146 120)(44 109 147 131)(45 140 148 118)(46 107 149 129)(47 138 150 116)(48 105 151 127)(49 136 152 114)(50 103 153 125)(51 134 154 112)(52 101 155 123)(53 132 156 110)(54 119 157 121)(55 130 158 108)(56 117 159 139)(57 128 160 106)(58 115 141 137)(59 126 142 104)(60 113 143 135)
G:=sub<Sym(160)| (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,85,31,95)(22,86,32,96)(23,87,33,97)(24,88,34,98)(25,89,35,99)(26,90,36,100)(27,91,37,81)(28,92,38,82)(29,93,39,83)(30,94,40,84)(41,159,51,149)(42,160,52,150)(43,141,53,151)(44,142,54,152)(45,143,55,153)(46,144,56,154)(47,145,57,155)(48,146,58,156)(49,147,59,157)(50,148,60,158)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140), (1,118,63,140)(2,109,64,131)(3,120,65,122)(4,111,66,133)(5,102,67,124)(6,113,68,135)(7,104,69,126)(8,115,70,137)(9,106,71,128)(10,117,72,139)(11,108,73,130)(12,119,74,121)(13,110,75,132)(14,101,76,123)(15,112,77,134)(16,103,78,125)(17,114,79,136)(18,105,80,127)(19,116,61,138)(20,107,62,129)(21,48,100,151)(22,59,81,142)(23,50,82,153)(24,41,83,144)(25,52,84,155)(26,43,85,146)(27,54,86,157)(28,45,87,148)(29,56,88,159)(30,47,89,150)(31,58,90,141)(32,49,91,152)(33,60,92,143)(34,51,93,154)(35,42,94,145)(36,53,95,156)(37,44,96,147)(38,55,97,158)(39,46,98,149)(40,57,99,160), (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), (1,38,63,97)(2,86,64,27)(3,36,65,95)(4,84,66,25)(5,34,67,93)(6,82,68,23)(7,32,69,91)(8,100,70,21)(9,30,71,89)(10,98,72,39)(11,28,73,87)(12,96,74,37)(13,26,75,85)(14,94,76,35)(15,24,77,83)(16,92,78,33)(17,22,79,81)(18,90,80,31)(19,40,61,99)(20,88,62,29)(41,124,144,102)(42,111,145,133)(43,122,146,120)(44,109,147,131)(45,140,148,118)(46,107,149,129)(47,138,150,116)(48,105,151,127)(49,136,152,114)(50,103,153,125)(51,134,154,112)(52,101,155,123)(53,132,156,110)(54,119,157,121)(55,130,158,108)(56,117,159,139)(57,128,160,106)(58,115,141,137)(59,126,142,104)(60,113,143,135)>;
G:=Group( (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,85,31,95)(22,86,32,96)(23,87,33,97)(24,88,34,98)(25,89,35,99)(26,90,36,100)(27,91,37,81)(28,92,38,82)(29,93,39,83)(30,94,40,84)(41,159,51,149)(42,160,52,150)(43,141,53,151)(44,142,54,152)(45,143,55,153)(46,144,56,154)(47,145,57,155)(48,146,58,156)(49,147,59,157)(50,148,60,158)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(101,106,111,116)(102,107,112,117)(103,108,113,118)(104,109,114,119)(105,110,115,120)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140), (1,118,63,140)(2,109,64,131)(3,120,65,122)(4,111,66,133)(5,102,67,124)(6,113,68,135)(7,104,69,126)(8,115,70,137)(9,106,71,128)(10,117,72,139)(11,108,73,130)(12,119,74,121)(13,110,75,132)(14,101,76,123)(15,112,77,134)(16,103,78,125)(17,114,79,136)(18,105,80,127)(19,116,61,138)(20,107,62,129)(21,48,100,151)(22,59,81,142)(23,50,82,153)(24,41,83,144)(25,52,84,155)(26,43,85,146)(27,54,86,157)(28,45,87,148)(29,56,88,159)(30,47,89,150)(31,58,90,141)(32,49,91,152)(33,60,92,143)(34,51,93,154)(35,42,94,145)(36,53,95,156)(37,44,96,147)(38,55,97,158)(39,46,98,149)(40,57,99,160), (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), (1,38,63,97)(2,86,64,27)(3,36,65,95)(4,84,66,25)(5,34,67,93)(6,82,68,23)(7,32,69,91)(8,100,70,21)(9,30,71,89)(10,98,72,39)(11,28,73,87)(12,96,74,37)(13,26,75,85)(14,94,76,35)(15,24,77,83)(16,92,78,33)(17,22,79,81)(18,90,80,31)(19,40,61,99)(20,88,62,29)(41,124,144,102)(42,111,145,133)(43,122,146,120)(44,109,147,131)(45,140,148,118)(46,107,149,129)(47,138,150,116)(48,105,151,127)(49,136,152,114)(50,103,153,125)(51,134,154,112)(52,101,155,123)(53,132,156,110)(54,119,157,121)(55,130,158,108)(56,117,159,139)(57,128,160,106)(58,115,141,137)(59,126,142,104)(60,113,143,135) );
G=PermutationGroup([(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,85,31,95),(22,86,32,96),(23,87,33,97),(24,88,34,98),(25,89,35,99),(26,90,36,100),(27,91,37,81),(28,92,38,82),(29,93,39,83),(30,94,40,84),(41,159,51,149),(42,160,52,150),(43,141,53,151),(44,142,54,152),(45,143,55,153),(46,144,56,154),(47,145,57,155),(48,146,58,156),(49,147,59,157),(50,148,60,158),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70),(101,106,111,116),(102,107,112,117),(103,108,113,118),(104,109,114,119),(105,110,115,120),(121,126,131,136),(122,127,132,137),(123,128,133,138),(124,129,134,139),(125,130,135,140)], [(1,118,63,140),(2,109,64,131),(3,120,65,122),(4,111,66,133),(5,102,67,124),(6,113,68,135),(7,104,69,126),(8,115,70,137),(9,106,71,128),(10,117,72,139),(11,108,73,130),(12,119,74,121),(13,110,75,132),(14,101,76,123),(15,112,77,134),(16,103,78,125),(17,114,79,136),(18,105,80,127),(19,116,61,138),(20,107,62,129),(21,48,100,151),(22,59,81,142),(23,50,82,153),(24,41,83,144),(25,52,84,155),(26,43,85,146),(27,54,86,157),(28,45,87,148),(29,56,88,159),(30,47,89,150),(31,58,90,141),(32,49,91,152),(33,60,92,143),(34,51,93,154),(35,42,94,145),(36,53,95,156),(37,44,96,147),(38,55,97,158),(39,46,98,149),(40,57,99,160)], [(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)], [(1,38,63,97),(2,86,64,27),(3,36,65,95),(4,84,66,25),(5,34,67,93),(6,82,68,23),(7,32,69,91),(8,100,70,21),(9,30,71,89),(10,98,72,39),(11,28,73,87),(12,96,74,37),(13,26,75,85),(14,94,76,35),(15,24,77,83),(16,92,78,33),(17,22,79,81),(18,90,80,31),(19,40,61,99),(20,88,62,29),(41,124,144,102),(42,111,145,133),(43,122,146,120),(44,109,147,131),(45,140,148,118),(46,107,149,129),(47,138,150,116),(48,105,151,127),(49,136,152,114),(50,103,153,125),(51,134,154,112),(52,101,155,123),(53,132,156,110),(54,119,157,121),(55,130,158,108),(56,117,159,139),(57,128,160,106),(58,115,141,137),(59,126,142,104),(60,113,143,135)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 38 | 40 | 0 | 0 |
| 0 | 0 | 35 | 6 | 30 | 28 |
| 0 | 0 | 19 | 32 | 22 | 11 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 23 | 10 | 7 |
| 0 | 0 | 6 | 30 | 10 | 33 |
| 0 | 0 | 2 | 21 | 36 | 40 |
| 0 | 0 | 36 | 38 | 8 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 16 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 35 | 34 | 27 | 25 |
| 0 | 0 | 28 | 10 | 5 | 32 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 26 | 37 | 19 |
| 0 | 0 | 34 | 20 | 10 | 31 |
| 0 | 0 | 17 | 13 | 27 | 9 |
| 0 | 0 | 7 | 6 | 17 | 29 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,35,19,0,0,28,40,6,32,0,0,0,0,30,22,0,0,0,0,28,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,6,2,36,0,0,23,30,21,38,0,0,10,10,36,8,0,0,7,33,40,15],[16,0,0,0,0,0,0,23,0,0,0,0,0,0,15,10,35,28,0,0,16,8,34,10,0,0,0,0,27,5,0,0,0,0,25,32],[0,25,0,0,0,0,18,0,0,0,0,0,0,0,6,34,17,7,0,0,26,20,13,6,0,0,37,10,27,17,0,0,19,31,9,29] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | D4.10D10 |
| kernel | C42.89D10 | C4×Dic10 | C20⋊2Q8 | C20.6Q8 | C23.D10 | Dic5.Q8 | C20.48D4 | C5×C42⋊C2 | C42⋊C2 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{89}D_{10} % in TeX
G:=Group("C4^2.89D10"); // GroupNames label
G:=SmallGroup(320,1190);
// by ID
G=gap.SmallGroup(320,1190);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀