metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.90D10, C10.482- (1+4), C10.922+ (1+4), (C2×C20)⋊5Q8, C20⋊Q8⋊11C2, C20⋊2Q8⋊6C2, (C2×C4)⋊4Dic10, C20.78(C2×Q8), C4⋊C4.268D10, (C4×C20).7C22, C20.6Q8⋊4C2, (C2×C10).63C24, C22⋊C4.91D10, C4.Dic10⋊11C2, C4.34(C2×Dic10), C2.6(D4⋊8D10), C10.11(C22×Q8), (C2×C20).142C23, C42⋊C2.13D5, (C22×C4).187D10, C4⋊Dic5.32C22, C22.7(C2×Dic10), C22.96(C23×D5), (C2×Dic5).22C23, (C4×Dic5).76C22, C2.13(C22×Dic10), C10.D4.2C22, C23.151(C22×D5), C2.7(D4.10D10), C23.D5.92C22, (C22×C20).223C22, (C22×C10).133C23, Dic5.14D4.1C2, C5⋊2(C23.41C23), (C2×Dic10).25C22, C23.21D10.23C2, (C22×Dic5).85C22, (C2×C10).13(C2×Q8), (C2×C4⋊Dic5).45C2, (C5×C4⋊C4).304C22, (C2×C4).148(C22×D5), (C5×C42⋊C2).14C2, (C5×C22⋊C4).99C22, SmallGroup(320,1191)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 638 in 206 conjugacy classes, 111 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.41C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], C4⋊Dic5 [×8], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C20⋊2Q8 [×2], C20.6Q8 [×2], Dic5.14D4 [×4], C20⋊Q8 [×2], C4.Dic10 [×2], C2×C4⋊Dic5, C23.21D10, C5×C42⋊C2, C42.90D10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ (1+4), 2- (1+4), Dic10 [×4], C22×D5 [×7], C23.41C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D4⋊8D10, D4.10D10, C42.90D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 103 16 88)(2 109 17 84)(3 105 18 90)(4 101 19 86)(5 107 20 82)(6 95 62 130)(7 91 63 126)(8 97 64 122)(9 93 65 128)(10 99 61 124)(11 83 77 108)(12 89 78 104)(13 85 79 110)(14 81 80 106)(15 87 76 102)(21 94 29 129)(22 100 30 125)(23 96 26 121)(24 92 27 127)(25 98 28 123)(31 114 39 149)(32 120 40 145)(33 116 36 141)(34 112 37 147)(35 118 38 143)(41 157 56 138)(42 153 57 134)(43 159 58 140)(44 155 59 136)(45 151 60 132)(46 152 51 133)(47 158 52 139)(48 154 53 135)(49 160 54 131)(50 156 55 137)(66 111 75 146)(67 117 71 142)(68 113 72 148)(69 119 73 144)(70 115 74 150)
(1 46 77 41)(2 47 78 42)(3 48 79 43)(4 49 80 44)(5 50 76 45)(6 70 22 32)(7 66 23 33)(8 67 24 34)(9 68 25 35)(10 69 21 31)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(26 36 63 75)(27 37 64 71)(28 38 65 72)(29 39 61 73)(30 40 62 74)(81 136 86 131)(82 137 87 132)(83 138 88 133)(84 139 89 134)(85 140 90 135)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(101 160 106 155)(102 151 107 156)(103 152 108 157)(104 153 109 158)(105 154 110 159)(121 141 126 146)(122 142 127 147)(123 143 128 148)(124 144 129 149)(125 145 130 150)
(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 26 11 7)(2 30 12 6)(3 29 13 10)(4 28 14 9)(5 27 15 8)(16 23 77 63)(17 22 78 62)(18 21 79 61)(19 25 80 65)(20 24 76 64)(31 48 73 58)(32 47 74 57)(33 46 75 56)(34 50 71 60)(35 49 72 59)(36 51 66 41)(37 55 67 45)(38 54 68 44)(39 53 69 43)(40 52 70 42)(81 128 101 98)(82 127 102 97)(83 126 103 96)(84 125 104 95)(85 124 105 94)(86 123 106 93)(87 122 107 92)(88 121 108 91)(89 130 109 100)(90 129 110 99)(111 138 141 152)(112 137 142 151)(113 136 143 160)(114 135 144 159)(115 134 145 158)(116 133 146 157)(117 132 147 156)(118 131 148 155)(119 140 149 154)(120 139 150 153)
G:=sub<Sym(160)| (1,103,16,88)(2,109,17,84)(3,105,18,90)(4,101,19,86)(5,107,20,82)(6,95,62,130)(7,91,63,126)(8,97,64,122)(9,93,65,128)(10,99,61,124)(11,83,77,108)(12,89,78,104)(13,85,79,110)(14,81,80,106)(15,87,76,102)(21,94,29,129)(22,100,30,125)(23,96,26,121)(24,92,27,127)(25,98,28,123)(31,114,39,149)(32,120,40,145)(33,116,36,141)(34,112,37,147)(35,118,38,143)(41,157,56,138)(42,153,57,134)(43,159,58,140)(44,155,59,136)(45,151,60,132)(46,152,51,133)(47,158,52,139)(48,154,53,135)(49,160,54,131)(50,156,55,137)(66,111,75,146)(67,117,71,142)(68,113,72,148)(69,119,73,144)(70,115,74,150), (1,46,77,41)(2,47,78,42)(3,48,79,43)(4,49,80,44)(5,50,76,45)(6,70,22,32)(7,66,23,33)(8,67,24,34)(9,68,25,35)(10,69,21,31)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(26,36,63,75)(27,37,64,71)(28,38,65,72)(29,39,61,73)(30,40,62,74)(81,136,86,131)(82,137,87,132)(83,138,88,133)(84,139,89,134)(85,140,90,135)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(101,160,106,155)(102,151,107,156)(103,152,108,157)(104,153,109,158)(105,154,110,159)(121,141,126,146)(122,142,127,147)(123,143,128,148)(124,144,129,149)(125,145,130,150), (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), (1,26,11,7)(2,30,12,6)(3,29,13,10)(4,28,14,9)(5,27,15,8)(16,23,77,63)(17,22,78,62)(18,21,79,61)(19,25,80,65)(20,24,76,64)(31,48,73,58)(32,47,74,57)(33,46,75,56)(34,50,71,60)(35,49,72,59)(36,51,66,41)(37,55,67,45)(38,54,68,44)(39,53,69,43)(40,52,70,42)(81,128,101,98)(82,127,102,97)(83,126,103,96)(84,125,104,95)(85,124,105,94)(86,123,106,93)(87,122,107,92)(88,121,108,91)(89,130,109,100)(90,129,110,99)(111,138,141,152)(112,137,142,151)(113,136,143,160)(114,135,144,159)(115,134,145,158)(116,133,146,157)(117,132,147,156)(118,131,148,155)(119,140,149,154)(120,139,150,153)>;
G:=Group( (1,103,16,88)(2,109,17,84)(3,105,18,90)(4,101,19,86)(5,107,20,82)(6,95,62,130)(7,91,63,126)(8,97,64,122)(9,93,65,128)(10,99,61,124)(11,83,77,108)(12,89,78,104)(13,85,79,110)(14,81,80,106)(15,87,76,102)(21,94,29,129)(22,100,30,125)(23,96,26,121)(24,92,27,127)(25,98,28,123)(31,114,39,149)(32,120,40,145)(33,116,36,141)(34,112,37,147)(35,118,38,143)(41,157,56,138)(42,153,57,134)(43,159,58,140)(44,155,59,136)(45,151,60,132)(46,152,51,133)(47,158,52,139)(48,154,53,135)(49,160,54,131)(50,156,55,137)(66,111,75,146)(67,117,71,142)(68,113,72,148)(69,119,73,144)(70,115,74,150), (1,46,77,41)(2,47,78,42)(3,48,79,43)(4,49,80,44)(5,50,76,45)(6,70,22,32)(7,66,23,33)(8,67,24,34)(9,68,25,35)(10,69,21,31)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(26,36,63,75)(27,37,64,71)(28,38,65,72)(29,39,61,73)(30,40,62,74)(81,136,86,131)(82,137,87,132)(83,138,88,133)(84,139,89,134)(85,140,90,135)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(101,160,106,155)(102,151,107,156)(103,152,108,157)(104,153,109,158)(105,154,110,159)(121,141,126,146)(122,142,127,147)(123,143,128,148)(124,144,129,149)(125,145,130,150), (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), (1,26,11,7)(2,30,12,6)(3,29,13,10)(4,28,14,9)(5,27,15,8)(16,23,77,63)(17,22,78,62)(18,21,79,61)(19,25,80,65)(20,24,76,64)(31,48,73,58)(32,47,74,57)(33,46,75,56)(34,50,71,60)(35,49,72,59)(36,51,66,41)(37,55,67,45)(38,54,68,44)(39,53,69,43)(40,52,70,42)(81,128,101,98)(82,127,102,97)(83,126,103,96)(84,125,104,95)(85,124,105,94)(86,123,106,93)(87,122,107,92)(88,121,108,91)(89,130,109,100)(90,129,110,99)(111,138,141,152)(112,137,142,151)(113,136,143,160)(114,135,144,159)(115,134,145,158)(116,133,146,157)(117,132,147,156)(118,131,148,155)(119,140,149,154)(120,139,150,153) );
G=PermutationGroup([(1,103,16,88),(2,109,17,84),(3,105,18,90),(4,101,19,86),(5,107,20,82),(6,95,62,130),(7,91,63,126),(8,97,64,122),(9,93,65,128),(10,99,61,124),(11,83,77,108),(12,89,78,104),(13,85,79,110),(14,81,80,106),(15,87,76,102),(21,94,29,129),(22,100,30,125),(23,96,26,121),(24,92,27,127),(25,98,28,123),(31,114,39,149),(32,120,40,145),(33,116,36,141),(34,112,37,147),(35,118,38,143),(41,157,56,138),(42,153,57,134),(43,159,58,140),(44,155,59,136),(45,151,60,132),(46,152,51,133),(47,158,52,139),(48,154,53,135),(49,160,54,131),(50,156,55,137),(66,111,75,146),(67,117,71,142),(68,113,72,148),(69,119,73,144),(70,115,74,150)], [(1,46,77,41),(2,47,78,42),(3,48,79,43),(4,49,80,44),(5,50,76,45),(6,70,22,32),(7,66,23,33),(8,67,24,34),(9,68,25,35),(10,69,21,31),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(26,36,63,75),(27,37,64,71),(28,38,65,72),(29,39,61,73),(30,40,62,74),(81,136,86,131),(82,137,87,132),(83,138,88,133),(84,139,89,134),(85,140,90,135),(91,111,96,116),(92,112,97,117),(93,113,98,118),(94,114,99,119),(95,115,100,120),(101,160,106,155),(102,151,107,156),(103,152,108,157),(104,153,109,158),(105,154,110,159),(121,141,126,146),(122,142,127,147),(123,143,128,148),(124,144,129,149),(125,145,130,150)], [(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)], [(1,26,11,7),(2,30,12,6),(3,29,13,10),(4,28,14,9),(5,27,15,8),(16,23,77,63),(17,22,78,62),(18,21,79,61),(19,25,80,65),(20,24,76,64),(31,48,73,58),(32,47,74,57),(33,46,75,56),(34,50,71,60),(35,49,72,59),(36,51,66,41),(37,55,67,45),(38,54,68,44),(39,53,69,43),(40,52,70,42),(81,128,101,98),(82,127,102,97),(83,126,103,96),(84,125,104,95),(85,124,105,94),(86,123,106,93),(87,122,107,92),(88,121,108,91),(89,130,109,100),(90,129,110,99),(111,138,141,152),(112,137,142,151),(113,136,143,160),(114,135,144,159),(115,134,145,158),(116,133,146,157),(117,132,147,156),(118,131,148,155),(119,140,149,154),(120,139,150,153)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 1 | 0 | 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 | 2 | 13 |
| 0 | 0 | 0 | 0 | 28 | 39 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 14 | 0 | 0 |
| 0 | 0 | 31 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 14 |
| 0 | 0 | 0 | 0 | 31 | 4 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,37,31,0,0,0,0,14,4,0,0,0,0,0,0,37,31,0,0,0,0,14,4] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | D10 | D10 | Dic10 | 2+ (1+4) | 2- (1+4) | D4⋊8D10 | D4.10D10 |
| kernel | C42.90D10 | C20⋊2Q8 | C20.6Q8 | Dic5.14D4 | C20⋊Q8 | C4.Dic10 | C2×C4⋊Dic5 | C23.21D10 | C5×C42⋊C2 | C2×C20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{90}D_{10} % in TeX
G:=Group("C4^2.90D10"); // GroupNames label
G:=SmallGroup(320,1191);
// by ID
G=gap.SmallGroup(320,1191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,184,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀