metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.134D10, C10.142- 1+4, (C4×Q8)⋊16D5, (Q8×C20)⋊18C2, C4⋊C4.301D10, (C4×Dic10)⋊40C2, C4.50(C4○D20), (C2×Q8).182D10, C4.Dic10⋊17C2, C20.6Q8⋊27C2, C42⋊D5.5C2, C42⋊2D5.2C2, Dic5⋊Q8⋊10C2, C20.121(C4○D4), (C2×C10).127C24, (C4×C20).179C22, (C2×C20).624C23, D10⋊3Q8.11C2, D10⋊2Q8.11C2, Dic5.Q8⋊10C2, C4⋊Dic5.370C22, (Q8×C10).227C22, (C2×Dic5).58C23, (C4×Dic5).95C22, (C22×D5).49C23, C22.148(C23×D5), C5⋊2(C22.35C24), C10.D4.78C22, C2.24(D4.10D10), D10⋊C4.126C22, C2.15(Q8.10D10), (C2×Dic10).300C22, C4⋊C4⋊D5.1C2, C10.57(C2×C4○D4), C2.66(C2×C4○D20), (C2×C4×D5).86C22, (C5×C4⋊C4).355C22, (C2×C4).290(C22×D5), SmallGroup(320,1255)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.134D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c9 >
Subgroups: 574 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.35C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Q8×C10, C4×Dic10, C20.6Q8, C42⋊D5, C42⋊2D5, Dic5.Q8, C4.Dic10, D10⋊2Q8, C4⋊C4⋊D5, Dic5⋊Q8, D10⋊3Q8, Q8×C20, C42.134D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.35C24, C4○D20, C23×D5, C2×C4○D20, Q8.10D10, D4.10D10, C42.134D10
►On 160 points
Generators in S160
(1 148 72 46)(2 47 73 149)(3 150 74 48)(4 49 75 151)(5 152 76 50)(6 51 77 153)(7 154 78 52)(8 53 79 155)(9 156 80 54)(10 55 61 157)(11 158 62 56)(12 57 63 159)(13 160 64 58)(14 59 65 141)(15 142 66 60)(16 41 67 143)(17 144 68 42)(18 43 69 145)(19 146 70 44)(20 45 71 147)(21 114 90 138)(22 139 91 115)(23 116 92 140)(24 121 93 117)(25 118 94 122)(26 123 95 119)(27 120 96 124)(28 125 97 101)(29 102 98 126)(30 127 99 103)(31 104 100 128)(32 129 81 105)(33 106 82 130)(34 131 83 107)(35 108 84 132)(36 133 85 109)(37 110 86 134)(38 135 87 111)(39 112 88 136)(40 137 89 113)
(1 101 62 135)(2 102 63 136)(3 103 64 137)(4 104 65 138)(5 105 66 139)(6 106 67 140)(7 107 68 121)(8 108 69 122)(9 109 70 123)(10 110 71 124)(11 111 72 125)(12 112 73 126)(13 113 74 127)(14 114 75 128)(15 115 76 129)(16 116 77 130)(17 117 78 131)(18 118 79 132)(19 119 80 133)(20 120 61 134)(21 49 100 141)(22 50 81 142)(23 51 82 143)(24 52 83 144)(25 53 84 145)(26 54 85 146)(27 55 86 147)(28 56 87 148)(29 57 88 149)(30 58 89 150)(31 59 90 151)(32 60 91 152)(33 41 92 153)(34 42 93 154)(35 43 94 155)(36 44 95 156)(37 45 96 157)(38 46 97 158)(39 47 98 159)(40 48 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 10 72 61)(2 80 73 9)(3 8 74 79)(4 78 75 7)(5 6 76 77)(11 20 62 71)(12 70 63 19)(13 18 64 69)(14 68 65 17)(15 16 66 67)(21 24 90 93)(22 92 91 23)(25 40 94 89)(26 88 95 39)(27 38 96 87)(28 86 97 37)(29 36 98 85)(30 84 99 35)(31 34 100 83)(32 82 81 33)(41 142 143 60)(42 59 144 141)(43 160 145 58)(44 57 146 159)(45 158 147 56)(46 55 148 157)(47 156 149 54)(48 53 150 155)(49 154 151 52)(50 51 152 153)(101 134 125 110)(102 109 126 133)(103 132 127 108)(104 107 128 131)(105 130 129 106)(111 124 135 120)(112 119 136 123)(113 122 137 118)(114 117 138 121)(115 140 139 116)
G:=sub<Sym(160)| (1,148,72,46)(2,47,73,149)(3,150,74,48)(4,49,75,151)(5,152,76,50)(6,51,77,153)(7,154,78,52)(8,53,79,155)(9,156,80,54)(10,55,61,157)(11,158,62,56)(12,57,63,159)(13,160,64,58)(14,59,65,141)(15,142,66,60)(16,41,67,143)(17,144,68,42)(18,43,69,145)(19,146,70,44)(20,45,71,147)(21,114,90,138)(22,139,91,115)(23,116,92,140)(24,121,93,117)(25,118,94,122)(26,123,95,119)(27,120,96,124)(28,125,97,101)(29,102,98,126)(30,127,99,103)(31,104,100,128)(32,129,81,105)(33,106,82,130)(34,131,83,107)(35,108,84,132)(36,133,85,109)(37,110,86,134)(38,135,87,111)(39,112,88,136)(40,137,89,113), (1,101,62,135)(2,102,63,136)(3,103,64,137)(4,104,65,138)(5,105,66,139)(6,106,67,140)(7,107,68,121)(8,108,69,122)(9,109,70,123)(10,110,71,124)(11,111,72,125)(12,112,73,126)(13,113,74,127)(14,114,75,128)(15,115,76,129)(16,116,77,130)(17,117,78,131)(18,118,79,132)(19,119,80,133)(20,120,61,134)(21,49,100,141)(22,50,81,142)(23,51,82,143)(24,52,83,144)(25,53,84,145)(26,54,85,146)(27,55,86,147)(28,56,87,148)(29,57,88,149)(30,58,89,150)(31,59,90,151)(32,60,91,152)(33,41,92,153)(34,42,93,154)(35,43,94,155)(36,44,95,156)(37,45,96,157)(38,46,97,158)(39,47,98,159)(40,48,99,160), (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), (1,10,72,61)(2,80,73,9)(3,8,74,79)(4,78,75,7)(5,6,76,77)(11,20,62,71)(12,70,63,19)(13,18,64,69)(14,68,65,17)(15,16,66,67)(21,24,90,93)(22,92,91,23)(25,40,94,89)(26,88,95,39)(27,38,96,87)(28,86,97,37)(29,36,98,85)(30,84,99,35)(31,34,100,83)(32,82,81,33)(41,142,143,60)(42,59,144,141)(43,160,145,58)(44,57,146,159)(45,158,147,56)(46,55,148,157)(47,156,149,54)(48,53,150,155)(49,154,151,52)(50,51,152,153)(101,134,125,110)(102,109,126,133)(103,132,127,108)(104,107,128,131)(105,130,129,106)(111,124,135,120)(112,119,136,123)(113,122,137,118)(114,117,138,121)(115,140,139,116)>;
G:=Group( (1,148,72,46)(2,47,73,149)(3,150,74,48)(4,49,75,151)(5,152,76,50)(6,51,77,153)(7,154,78,52)(8,53,79,155)(9,156,80,54)(10,55,61,157)(11,158,62,56)(12,57,63,159)(13,160,64,58)(14,59,65,141)(15,142,66,60)(16,41,67,143)(17,144,68,42)(18,43,69,145)(19,146,70,44)(20,45,71,147)(21,114,90,138)(22,139,91,115)(23,116,92,140)(24,121,93,117)(25,118,94,122)(26,123,95,119)(27,120,96,124)(28,125,97,101)(29,102,98,126)(30,127,99,103)(31,104,100,128)(32,129,81,105)(33,106,82,130)(34,131,83,107)(35,108,84,132)(36,133,85,109)(37,110,86,134)(38,135,87,111)(39,112,88,136)(40,137,89,113), (1,101,62,135)(2,102,63,136)(3,103,64,137)(4,104,65,138)(5,105,66,139)(6,106,67,140)(7,107,68,121)(8,108,69,122)(9,109,70,123)(10,110,71,124)(11,111,72,125)(12,112,73,126)(13,113,74,127)(14,114,75,128)(15,115,76,129)(16,116,77,130)(17,117,78,131)(18,118,79,132)(19,119,80,133)(20,120,61,134)(21,49,100,141)(22,50,81,142)(23,51,82,143)(24,52,83,144)(25,53,84,145)(26,54,85,146)(27,55,86,147)(28,56,87,148)(29,57,88,149)(30,58,89,150)(31,59,90,151)(32,60,91,152)(33,41,92,153)(34,42,93,154)(35,43,94,155)(36,44,95,156)(37,45,96,157)(38,46,97,158)(39,47,98,159)(40,48,99,160), (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), (1,10,72,61)(2,80,73,9)(3,8,74,79)(4,78,75,7)(5,6,76,77)(11,20,62,71)(12,70,63,19)(13,18,64,69)(14,68,65,17)(15,16,66,67)(21,24,90,93)(22,92,91,23)(25,40,94,89)(26,88,95,39)(27,38,96,87)(28,86,97,37)(29,36,98,85)(30,84,99,35)(31,34,100,83)(32,82,81,33)(41,142,143,60)(42,59,144,141)(43,160,145,58)(44,57,146,159)(45,158,147,56)(46,55,148,157)(47,156,149,54)(48,53,150,155)(49,154,151,52)(50,51,152,153)(101,134,125,110)(102,109,126,133)(103,132,127,108)(104,107,128,131)(105,130,129,106)(111,124,135,120)(112,119,136,123)(113,122,137,118)(114,117,138,121)(115,140,139,116) );
G=PermutationGroup([[(1,148,72,46),(2,47,73,149),(3,150,74,48),(4,49,75,151),(5,152,76,50),(6,51,77,153),(7,154,78,52),(8,53,79,155),(9,156,80,54),(10,55,61,157),(11,158,62,56),(12,57,63,159),(13,160,64,58),(14,59,65,141),(15,142,66,60),(16,41,67,143),(17,144,68,42),(18,43,69,145),(19,146,70,44),(20,45,71,147),(21,114,90,138),(22,139,91,115),(23,116,92,140),(24,121,93,117),(25,118,94,122),(26,123,95,119),(27,120,96,124),(28,125,97,101),(29,102,98,126),(30,127,99,103),(31,104,100,128),(32,129,81,105),(33,106,82,130),(34,131,83,107),(35,108,84,132),(36,133,85,109),(37,110,86,134),(38,135,87,111),(39,112,88,136),(40,137,89,113)], [(1,101,62,135),(2,102,63,136),(3,103,64,137),(4,104,65,138),(5,105,66,139),(6,106,67,140),(7,107,68,121),(8,108,69,122),(9,109,70,123),(10,110,71,124),(11,111,72,125),(12,112,73,126),(13,113,74,127),(14,114,75,128),(15,115,76,129),(16,116,77,130),(17,117,78,131),(18,118,79,132),(19,119,80,133),(20,120,61,134),(21,49,100,141),(22,50,81,142),(23,51,82,143),(24,52,83,144),(25,53,84,145),(26,54,85,146),(27,55,86,147),(28,56,87,148),(29,57,88,149),(30,58,89,150),(31,59,90,151),(32,60,91,152),(33,41,92,153),(34,42,93,154),(35,43,94,155),(36,44,95,156),(37,45,96,157),(38,46,97,158),(39,47,98,159),(40,48,99,160)], [(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)], [(1,10,72,61),(2,80,73,9),(3,8,74,79),(4,78,75,7),(5,6,76,77),(11,20,62,71),(12,70,63,19),(13,18,64,69),(14,68,65,17),(15,16,66,67),(21,24,90,93),(22,92,91,23),(25,40,94,89),(26,88,95,39),(27,38,96,87),(28,86,97,37),(29,36,98,85),(30,84,99,35),(31,34,100,83),(32,82,81,33),(41,142,143,60),(42,59,144,141),(43,160,145,58),(44,57,146,159),(45,158,147,56),(46,55,148,157),(47,156,149,54),(48,53,150,155),(49,154,151,52),(50,51,152,153),(101,134,125,110),(102,109,126,133),(103,132,127,108),(104,107,128,131),(105,130,129,106),(111,124,135,120),(112,119,136,123),(113,122,137,118),(114,117,138,121),(115,140,139,116)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | Q8.10D10 | D4.10D10 |
| kernel | C42.134D10 | C4×Dic10 | C20.6Q8 | C42⋊D5 | C42⋊2D5 | Dic5.Q8 | C4.Dic10 | D10⋊2Q8 | C4⋊C4⋊D5 | Dic5⋊Q8 | D10⋊3Q8 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 2 | 16 | 2 | 4 | 4 |
Matrix representation of C42.134D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 14 | 11 |
| 0 | 0 | 0 | 31 | 14 | 14 |
| 0 | 0 | 21 | 4 | 10 | 0 |
| 0 | 0 | 20 | 21 | 0 | 10 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 15 | 37 | 0 | 0 | 0 | 0 |
| 36 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 32 | 11 |
| 0 | 0 | 34 | 40 | 11 | 32 |
| 0 | 0 | 19 | 28 | 1 | 34 |
| 0 | 0 | 23 | 19 | 7 | 34 |
| 26 | 4 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 32 | 11 |
| 0 | 0 | 40 | 34 | 11 | 14 |
| 0 | 0 | 28 | 19 | 1 | 0 |
| 0 | 0 | 19 | 23 | 7 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,0,21,20,0,0,0,31,4,21,0,0,14,14,10,0,0,0,11,14,0,10],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[15,36,0,0,0,0,37,26,0,0,0,0,0,0,7,34,19,23,0,0,7,40,28,19,0,0,32,11,1,7,0,0,11,32,34,34],[26,26,0,0,0,0,4,15,0,0,0,0,0,0,7,40,28,19,0,0,7,34,19,23,0,0,32,11,1,7,0,0,11,14,0,40] >;
C42.134D10 in GAP, Magma, Sage, TeX
C_4^2._{134}D_{10} % in TeX
G:=Group("C4^2.134D10"); // GroupNames label
G:=SmallGroup(320,1255);
// by ID
G=gap.SmallGroup(320,1255);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations