metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.171D10, C10.342- 1+4, C4⋊Q8⋊9D5, C4.37(D4×D5), (C4×D5).13D4, C20.69(C2×D4), C4⋊C4.122D10, D10.82(C2×D4), D10⋊Q8⋊47C2, C4.D20⋊26C2, C42⋊D5⋊25C2, (C2×Q8).143D10, Dic5.93(C2×D4), C10.98(C22×D4), Dic5⋊Q8⋊26C2, C20.23D4⋊25C2, (C2×C10).268C24, (C4×C20).209C22, (C2×C20).101C23, D10.13D4⋊45C2, (C2×D20).177C22, (Q8×C10).135C22, C22.289(C23×D5), D10⋊C4.49C22, C5⋊5(C23.38C23), (C2×Dic5).140C23, (C4×Dic5).167C22, (C22×D5).240C23, C2.35(Q8.10D10), (C2×Dic10).194C22, C10.D4.165C22, (C2×Q8×D5)⋊12C2, C2.71(C2×D4×D5), (C5×C4⋊Q8)⋊10C2, (C2×Q8⋊2D5).7C2, (C2×C4×D5).151C22, (C5×C4⋊C4).211C22, (C2×C4).217(C22×D5), SmallGroup(320,1396)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.171D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 990 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C23.38C23, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8×D5, Q8⋊2D5, Q8×C10, C42⋊D5, C4.D20, D10.13D4, D10⋊Q8, Dic5⋊Q8, C20.23D4, C5×C4⋊Q8, C2×Q8×D5, C2×Q8⋊2D5, C42.171D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2- 1+4, C22×D5, C23.38C23, D4×D5, C23×D5, C2×D4×D5, Q8.10D10, C42.171D10
►On 160 points
Generators in S160
(1 59 11 49)(2 50 12 60)(3 41 13 51)(4 52 14 42)(5 43 15 53)(6 54 16 44)(7 45 17 55)(8 56 18 46)(9 47 19 57)(10 58 20 48)(21 64 31 74)(22 75 32 65)(23 66 33 76)(24 77 34 67)(25 68 35 78)(26 79 36 69)(27 70 37 80)(28 61 38 71)(29 72 39 62)(30 63 40 73)(81 116 91 106)(82 107 92 117)(83 118 93 108)(84 109 94 119)(85 120 95 110)(86 111 96 101)(87 102 97 112)(88 113 98 103)(89 104 99 114)(90 115 100 105)(121 155 131 145)(122 146 132 156)(123 157 133 147)(124 148 134 158)(125 159 135 149)(126 150 136 160)(127 141 137 151)(128 152 138 142)(129 143 139 153)(130 154 140 144)
(1 104 40 125)(2 126 21 105)(3 106 22 127)(4 128 23 107)(5 108 24 129)(6 130 25 109)(7 110 26 131)(8 132 27 111)(9 112 28 133)(10 134 29 113)(11 114 30 135)(12 136 31 115)(13 116 32 137)(14 138 33 117)(15 118 34 139)(16 140 35 119)(17 120 36 121)(18 122 37 101)(19 102 38 123)(20 124 39 103)(41 81 75 141)(42 142 76 82)(43 83 77 143)(44 144 78 84)(45 85 79 145)(46 146 80 86)(47 87 61 147)(48 148 62 88)(49 89 63 149)(50 150 64 90)(51 91 65 151)(52 152 66 92)(53 93 67 153)(54 154 68 94)(55 95 69 155)(56 156 70 96)(57 97 71 157)(58 158 72 98)(59 99 73 159)(60 160 74 100)
(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)
(1 129 11 139)(2 138 12 128)(3 127 13 137)(4 136 14 126)(5 125 15 135)(6 134 16 124)(7 123 17 133)(8 132 18 122)(9 121 19 131)(10 130 20 140)(21 117 31 107)(22 106 32 116)(23 115 33 105)(24 104 34 114)(25 113 35 103)(26 102 36 112)(27 111 37 101)(28 120 38 110)(29 109 39 119)(30 118 40 108)(41 151 51 141)(42 160 52 150)(43 149 53 159)(44 158 54 148)(45 147 55 157)(46 156 56 146)(47 145 57 155)(48 154 58 144)(49 143 59 153)(50 152 60 142)(61 85 71 95)(62 94 72 84)(63 83 73 93)(64 92 74 82)(65 81 75 91)(66 90 76 100)(67 99 77 89)(68 88 78 98)(69 97 79 87)(70 86 80 96)
G:=sub<Sym(160)| (1,59,11,49)(2,50,12,60)(3,41,13,51)(4,52,14,42)(5,43,15,53)(6,54,16,44)(7,45,17,55)(8,56,18,46)(9,47,19,57)(10,58,20,48)(21,64,31,74)(22,75,32,65)(23,66,33,76)(24,77,34,67)(25,68,35,78)(26,79,36,69)(27,70,37,80)(28,61,38,71)(29,72,39,62)(30,63,40,73)(81,116,91,106)(82,107,92,117)(83,118,93,108)(84,109,94,119)(85,120,95,110)(86,111,96,101)(87,102,97,112)(88,113,98,103)(89,104,99,114)(90,115,100,105)(121,155,131,145)(122,146,132,156)(123,157,133,147)(124,148,134,158)(125,159,135,149)(126,150,136,160)(127,141,137,151)(128,152,138,142)(129,143,139,153)(130,154,140,144), (1,104,40,125)(2,126,21,105)(3,106,22,127)(4,128,23,107)(5,108,24,129)(6,130,25,109)(7,110,26,131)(8,132,27,111)(9,112,28,133)(10,134,29,113)(11,114,30,135)(12,136,31,115)(13,116,32,137)(14,138,33,117)(15,118,34,139)(16,140,35,119)(17,120,36,121)(18,122,37,101)(19,102,38,123)(20,124,39,103)(41,81,75,141)(42,142,76,82)(43,83,77,143)(44,144,78,84)(45,85,79,145)(46,146,80,86)(47,87,61,147)(48,148,62,88)(49,89,63,149)(50,150,64,90)(51,91,65,151)(52,152,66,92)(53,93,67,153)(54,154,68,94)(55,95,69,155)(56,156,70,96)(57,97,71,157)(58,158,72,98)(59,99,73,159)(60,160,74,100), (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), (1,129,11,139)(2,138,12,128)(3,127,13,137)(4,136,14,126)(5,125,15,135)(6,134,16,124)(7,123,17,133)(8,132,18,122)(9,121,19,131)(10,130,20,140)(21,117,31,107)(22,106,32,116)(23,115,33,105)(24,104,34,114)(25,113,35,103)(26,102,36,112)(27,111,37,101)(28,120,38,110)(29,109,39,119)(30,118,40,108)(41,151,51,141)(42,160,52,150)(43,149,53,159)(44,158,54,148)(45,147,55,157)(46,156,56,146)(47,145,57,155)(48,154,58,144)(49,143,59,153)(50,152,60,142)(61,85,71,95)(62,94,72,84)(63,83,73,93)(64,92,74,82)(65,81,75,91)(66,90,76,100)(67,99,77,89)(68,88,78,98)(69,97,79,87)(70,86,80,96)>;
G:=Group( (1,59,11,49)(2,50,12,60)(3,41,13,51)(4,52,14,42)(5,43,15,53)(6,54,16,44)(7,45,17,55)(8,56,18,46)(9,47,19,57)(10,58,20,48)(21,64,31,74)(22,75,32,65)(23,66,33,76)(24,77,34,67)(25,68,35,78)(26,79,36,69)(27,70,37,80)(28,61,38,71)(29,72,39,62)(30,63,40,73)(81,116,91,106)(82,107,92,117)(83,118,93,108)(84,109,94,119)(85,120,95,110)(86,111,96,101)(87,102,97,112)(88,113,98,103)(89,104,99,114)(90,115,100,105)(121,155,131,145)(122,146,132,156)(123,157,133,147)(124,148,134,158)(125,159,135,149)(126,150,136,160)(127,141,137,151)(128,152,138,142)(129,143,139,153)(130,154,140,144), (1,104,40,125)(2,126,21,105)(3,106,22,127)(4,128,23,107)(5,108,24,129)(6,130,25,109)(7,110,26,131)(8,132,27,111)(9,112,28,133)(10,134,29,113)(11,114,30,135)(12,136,31,115)(13,116,32,137)(14,138,33,117)(15,118,34,139)(16,140,35,119)(17,120,36,121)(18,122,37,101)(19,102,38,123)(20,124,39,103)(41,81,75,141)(42,142,76,82)(43,83,77,143)(44,144,78,84)(45,85,79,145)(46,146,80,86)(47,87,61,147)(48,148,62,88)(49,89,63,149)(50,150,64,90)(51,91,65,151)(52,152,66,92)(53,93,67,153)(54,154,68,94)(55,95,69,155)(56,156,70,96)(57,97,71,157)(58,158,72,98)(59,99,73,159)(60,160,74,100), (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), (1,129,11,139)(2,138,12,128)(3,127,13,137)(4,136,14,126)(5,125,15,135)(6,134,16,124)(7,123,17,133)(8,132,18,122)(9,121,19,131)(10,130,20,140)(21,117,31,107)(22,106,32,116)(23,115,33,105)(24,104,34,114)(25,113,35,103)(26,102,36,112)(27,111,37,101)(28,120,38,110)(29,109,39,119)(30,118,40,108)(41,151,51,141)(42,160,52,150)(43,149,53,159)(44,158,54,148)(45,147,55,157)(46,156,56,146)(47,145,57,155)(48,154,58,144)(49,143,59,153)(50,152,60,142)(61,85,71,95)(62,94,72,84)(63,83,73,93)(64,92,74,82)(65,81,75,91)(66,90,76,100)(67,99,77,89)(68,88,78,98)(69,97,79,87)(70,86,80,96) );
G=PermutationGroup([[(1,59,11,49),(2,50,12,60),(3,41,13,51),(4,52,14,42),(5,43,15,53),(6,54,16,44),(7,45,17,55),(8,56,18,46),(9,47,19,57),(10,58,20,48),(21,64,31,74),(22,75,32,65),(23,66,33,76),(24,77,34,67),(25,68,35,78),(26,79,36,69),(27,70,37,80),(28,61,38,71),(29,72,39,62),(30,63,40,73),(81,116,91,106),(82,107,92,117),(83,118,93,108),(84,109,94,119),(85,120,95,110),(86,111,96,101),(87,102,97,112),(88,113,98,103),(89,104,99,114),(90,115,100,105),(121,155,131,145),(122,146,132,156),(123,157,133,147),(124,148,134,158),(125,159,135,149),(126,150,136,160),(127,141,137,151),(128,152,138,142),(129,143,139,153),(130,154,140,144)], [(1,104,40,125),(2,126,21,105),(3,106,22,127),(4,128,23,107),(5,108,24,129),(6,130,25,109),(7,110,26,131),(8,132,27,111),(9,112,28,133),(10,134,29,113),(11,114,30,135),(12,136,31,115),(13,116,32,137),(14,138,33,117),(15,118,34,139),(16,140,35,119),(17,120,36,121),(18,122,37,101),(19,102,38,123),(20,124,39,103),(41,81,75,141),(42,142,76,82),(43,83,77,143),(44,144,78,84),(45,85,79,145),(46,146,80,86),(47,87,61,147),(48,148,62,88),(49,89,63,149),(50,150,64,90),(51,91,65,151),(52,152,66,92),(53,93,67,153),(54,154,68,94),(55,95,69,155),(56,156,70,96),(57,97,71,157),(58,158,72,98),(59,99,73,159),(60,160,74,100)], [(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)], [(1,129,11,139),(2,138,12,128),(3,127,13,137),(4,136,14,126),(5,125,15,135),(6,134,16,124),(7,123,17,133),(8,132,18,122),(9,121,19,131),(10,130,20,140),(21,117,31,107),(22,106,32,116),(23,115,33,105),(24,104,34,114),(25,113,35,103),(26,102,36,112),(27,111,37,101),(28,120,38,110),(29,109,39,119),(30,118,40,108),(41,151,51,141),(42,160,52,150),(43,149,53,159),(44,158,54,148),(45,147,55,157),(46,156,56,146),(47,145,57,155),(48,154,58,144),(49,143,59,153),(50,152,60,142),(61,85,71,95),(62,94,72,84),(63,83,73,93),(64,92,74,82),(65,81,75,91),(66,90,76,100),(67,99,77,89),(68,88,78,98),(69,97,79,87),(70,86,80,96)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 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 | D4 | D5 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | Q8.10D10 |
| kernel | C42.171D10 | C42⋊D5 | C4.D20 | D10.13D4 | D10⋊Q8 | Dic5⋊Q8 | C20.23D4 | C5×C4⋊Q8 | C2×Q8×D5 | C2×Q8⋊2D5 | C4×D5 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 2 | 4 | 8 |
Matrix representation of C42.171D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 4 | 39 | 0 | 0 |
| 0 | 0 | 6 | 1 | 30 | 32 |
| 0 | 0 | 40 | 0 | 9 | 11 |
| 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 3 | 18 | 9 | 0 |
| 0 | 0 | 3 | 18 | 0 | 9 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 18 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 19 | 27 | 33 |
| 0 | 0 | 32 | 28 | 35 | 34 |
| 0 | 0 | 34 | 0 | 22 | 22 |
| 0 | 0 | 34 | 34 | 22 | 22 |
| 32 | 9 | 0 | 0 | 0 | 0 |
| 23 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 22 | 0 | 0 |
| 0 | 0 | 37 | 13 | 0 | 0 |
| 0 | 0 | 34 | 0 | 22 | 22 |
| 0 | 0 | 1 | 7 | 32 | 19 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,4,6,40,0,0,9,39,1,0,0,0,0,0,30,9,0,0,0,0,32,11],[32,0,0,0,0,0,9,9,0,0,0,0,0,0,32,0,3,3,0,0,0,32,18,18,0,0,0,0,9,0,0,0,0,0,0,9],[9,18,0,0,0,0,32,32,0,0,0,0,0,0,10,32,34,34,0,0,19,28,0,34,0,0,27,35,22,22,0,0,33,34,22,22],[32,23,0,0,0,0,9,9,0,0,0,0,0,0,28,37,34,1,0,0,22,13,0,7,0,0,0,0,22,32,0,0,0,0,22,19] >;
C42.171D10 in GAP, Magma, Sage, TeX
C_4^2._{171}D_{10} % in TeX
G:=Group("C4^2.171D10"); // GroupNames label
G:=SmallGroup(320,1396);
// by ID
G=gap.SmallGroup(320,1396);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,136,12550]);
// 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=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