metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.95D10, C10.972+ 1+4, (C4×D20)⋊10C2, D10⋊D4⋊3C2, C20⋊7D4⋊29C2, C4⋊D20⋊12C2, C4⋊C4.272D10, C42⋊2D5⋊4C2, D20⋊8C4⋊12C2, C4.96(C4○D20), C42⋊C2⋊14D5, (C2×C10).74C24, (C4×C20).25C22, C22⋊C4.98D10, C4.Dic10⋊13C2, D10.51(C4○D4), D10.12D4⋊4C2, C20.198(C4○D4), C2.9(D4⋊8D10), (C2×C20).149C23, (C22×C4).195D10, C23.86(C22×D5), D10⋊C4.3C22, (C2×D20).143C22, C4⋊Dic5.195C22, (C2×Dic5).27C23, (C4×Dic5).78C22, (C22×D5).22C23, C22.103(C23×D5), C23.D5.97C22, (C22×C20).232C22, (C22×C10).144C23, C5⋊2(C22.47C24), C10.D4.98C22, (D5×C4⋊C4)⋊13C2, (C4×C5⋊D4)⋊12C2, C2.13(D5×C4○D4), C2.33(C2×C4○D20), (C2×C4×D5).70C22, C10.134(C2×C4○D4), (C5×C42⋊C2)⋊16C2, (C5×C4⋊C4).310C22, (C2×C4).276(C22×D5), (C2×C5⋊D4).112C22, (C5×C22⋊C4).114C22, SmallGroup(320,1202)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.95D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 902 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.47C24, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C4×D20, C42⋊2D5, D10.12D4, D10⋊D4, C4.Dic10, D5×C4⋊C4, D20⋊8C4, C4⋊D20, C4×C5⋊D4, C20⋊7D4, C5×C42⋊C2, C42.95D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C4○D20, C23×D5, C2×C4○D20, D5×C4○D4, D4⋊8D10, C42.95D10
►On 160 points
Generators in S160
(1 119 95 149)(2 110 96 160)(3 101 97 151)(4 112 98 142)(5 103 99 153)(6 114 100 144)(7 105 81 155)(8 116 82 146)(9 107 83 157)(10 118 84 148)(11 109 85 159)(12 120 86 150)(13 111 87 141)(14 102 88 152)(15 113 89 143)(16 104 90 154)(17 115 91 145)(18 106 92 156)(19 117 93 147)(20 108 94 158)(21 66 50 135)(22 77 51 126)(23 68 52 137)(24 79 53 128)(25 70 54 139)(26 61 55 130)(27 72 56 121)(28 63 57 132)(29 74 58 123)(30 65 59 134)(31 76 60 125)(32 67 41 136)(33 78 42 127)(34 69 43 138)(35 80 44 129)(36 71 45 140)(37 62 46 131)(38 73 47 122)(39 64 48 133)(40 75 49 124)
(1 44 11 54)(2 45 12 55)(3 46 13 56)(4 47 14 57)(5 48 15 58)(6 49 16 59)(7 50 17 60)(8 51 18 41)(9 52 19 42)(10 53 20 43)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(61 160 71 150)(62 141 72 151)(63 142 73 152)(64 143 74 153)(65 144 75 154)(66 145 76 155)(67 146 77 156)(68 147 78 157)(69 148 79 158)(70 149 80 159)(101 131 111 121)(102 132 112 122)(103 133 113 123)(104 134 114 124)(105 135 115 125)(106 136 116 126)(107 137 117 127)(108 138 118 128)(109 139 119 129)(110 140 120 130)
(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 79 11 69)(2 68 12 78)(3 77 13 67)(4 66 14 76)(5 75 15 65)(6 64 16 74)(7 73 17 63)(8 62 18 72)(9 71 19 61)(10 80 20 70)(21 142 31 152)(22 151 32 141)(23 160 33 150)(24 149 34 159)(25 158 35 148)(26 147 36 157)(27 156 37 146)(28 145 38 155)(29 154 39 144)(30 143 40 153)(41 111 51 101)(42 120 52 110)(43 109 53 119)(44 118 54 108)(45 107 55 117)(46 116 56 106)(47 105 57 115)(48 114 58 104)(49 103 59 113)(50 112 60 102)(81 122 91 132)(82 131 92 121)(83 140 93 130)(84 129 94 139)(85 138 95 128)(86 127 96 137)(87 136 97 126)(88 125 98 135)(89 134 99 124)(90 123 100 133)
G:=sub<Sym(160)| (1,119,95,149)(2,110,96,160)(3,101,97,151)(4,112,98,142)(5,103,99,153)(6,114,100,144)(7,105,81,155)(8,116,82,146)(9,107,83,157)(10,118,84,148)(11,109,85,159)(12,120,86,150)(13,111,87,141)(14,102,88,152)(15,113,89,143)(16,104,90,154)(17,115,91,145)(18,106,92,156)(19,117,93,147)(20,108,94,158)(21,66,50,135)(22,77,51,126)(23,68,52,137)(24,79,53,128)(25,70,54,139)(26,61,55,130)(27,72,56,121)(28,63,57,132)(29,74,58,123)(30,65,59,134)(31,76,60,125)(32,67,41,136)(33,78,42,127)(34,69,43,138)(35,80,44,129)(36,71,45,140)(37,62,46,131)(38,73,47,122)(39,64,48,133)(40,75,49,124), (1,44,11,54)(2,45,12,55)(3,46,13,56)(4,47,14,57)(5,48,15,58)(6,49,16,59)(7,50,17,60)(8,51,18,41)(9,52,19,42)(10,53,20,43)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(61,160,71,150)(62,141,72,151)(63,142,73,152)(64,143,74,153)(65,144,75,154)(66,145,76,155)(67,146,77,156)(68,147,78,157)(69,148,79,158)(70,149,80,159)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125)(106,136,116,126)(107,137,117,127)(108,138,118,128)(109,139,119,129)(110,140,120,130), (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), (1,79,11,69)(2,68,12,78)(3,77,13,67)(4,66,14,76)(5,75,15,65)(6,64,16,74)(7,73,17,63)(8,62,18,72)(9,71,19,61)(10,80,20,70)(21,142,31,152)(22,151,32,141)(23,160,33,150)(24,149,34,159)(25,158,35,148)(26,147,36,157)(27,156,37,146)(28,145,38,155)(29,154,39,144)(30,143,40,153)(41,111,51,101)(42,120,52,110)(43,109,53,119)(44,118,54,108)(45,107,55,117)(46,116,56,106)(47,105,57,115)(48,114,58,104)(49,103,59,113)(50,112,60,102)(81,122,91,132)(82,131,92,121)(83,140,93,130)(84,129,94,139)(85,138,95,128)(86,127,96,137)(87,136,97,126)(88,125,98,135)(89,134,99,124)(90,123,100,133)>;
G:=Group( (1,119,95,149)(2,110,96,160)(3,101,97,151)(4,112,98,142)(5,103,99,153)(6,114,100,144)(7,105,81,155)(8,116,82,146)(9,107,83,157)(10,118,84,148)(11,109,85,159)(12,120,86,150)(13,111,87,141)(14,102,88,152)(15,113,89,143)(16,104,90,154)(17,115,91,145)(18,106,92,156)(19,117,93,147)(20,108,94,158)(21,66,50,135)(22,77,51,126)(23,68,52,137)(24,79,53,128)(25,70,54,139)(26,61,55,130)(27,72,56,121)(28,63,57,132)(29,74,58,123)(30,65,59,134)(31,76,60,125)(32,67,41,136)(33,78,42,127)(34,69,43,138)(35,80,44,129)(36,71,45,140)(37,62,46,131)(38,73,47,122)(39,64,48,133)(40,75,49,124), (1,44,11,54)(2,45,12,55)(3,46,13,56)(4,47,14,57)(5,48,15,58)(6,49,16,59)(7,50,17,60)(8,51,18,41)(9,52,19,42)(10,53,20,43)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(61,160,71,150)(62,141,72,151)(63,142,73,152)(64,143,74,153)(65,144,75,154)(66,145,76,155)(67,146,77,156)(68,147,78,157)(69,148,79,158)(70,149,80,159)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125)(106,136,116,126)(107,137,117,127)(108,138,118,128)(109,139,119,129)(110,140,120,130), (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), (1,79,11,69)(2,68,12,78)(3,77,13,67)(4,66,14,76)(5,75,15,65)(6,64,16,74)(7,73,17,63)(8,62,18,72)(9,71,19,61)(10,80,20,70)(21,142,31,152)(22,151,32,141)(23,160,33,150)(24,149,34,159)(25,158,35,148)(26,147,36,157)(27,156,37,146)(28,145,38,155)(29,154,39,144)(30,143,40,153)(41,111,51,101)(42,120,52,110)(43,109,53,119)(44,118,54,108)(45,107,55,117)(46,116,56,106)(47,105,57,115)(48,114,58,104)(49,103,59,113)(50,112,60,102)(81,122,91,132)(82,131,92,121)(83,140,93,130)(84,129,94,139)(85,138,95,128)(86,127,96,137)(87,136,97,126)(88,125,98,135)(89,134,99,124)(90,123,100,133) );
G=PermutationGroup([[(1,119,95,149),(2,110,96,160),(3,101,97,151),(4,112,98,142),(5,103,99,153),(6,114,100,144),(7,105,81,155),(8,116,82,146),(9,107,83,157),(10,118,84,148),(11,109,85,159),(12,120,86,150),(13,111,87,141),(14,102,88,152),(15,113,89,143),(16,104,90,154),(17,115,91,145),(18,106,92,156),(19,117,93,147),(20,108,94,158),(21,66,50,135),(22,77,51,126),(23,68,52,137),(24,79,53,128),(25,70,54,139),(26,61,55,130),(27,72,56,121),(28,63,57,132),(29,74,58,123),(30,65,59,134),(31,76,60,125),(32,67,41,136),(33,78,42,127),(34,69,43,138),(35,80,44,129),(36,71,45,140),(37,62,46,131),(38,73,47,122),(39,64,48,133),(40,75,49,124)], [(1,44,11,54),(2,45,12,55),(3,46,13,56),(4,47,14,57),(5,48,15,58),(6,49,16,59),(7,50,17,60),(8,51,18,41),(9,52,19,42),(10,53,20,43),(21,91,31,81),(22,92,32,82),(23,93,33,83),(24,94,34,84),(25,95,35,85),(26,96,36,86),(27,97,37,87),(28,98,38,88),(29,99,39,89),(30,100,40,90),(61,160,71,150),(62,141,72,151),(63,142,73,152),(64,143,74,153),(65,144,75,154),(66,145,76,155),(67,146,77,156),(68,147,78,157),(69,148,79,158),(70,149,80,159),(101,131,111,121),(102,132,112,122),(103,133,113,123),(104,134,114,124),(105,135,115,125),(106,136,116,126),(107,137,117,127),(108,138,118,128),(109,139,119,129),(110,140,120,130)], [(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)], [(1,79,11,69),(2,68,12,78),(3,77,13,67),(4,66,14,76),(5,75,15,65),(6,64,16,74),(7,73,17,63),(8,62,18,72),(9,71,19,61),(10,80,20,70),(21,142,31,152),(22,151,32,141),(23,160,33,150),(24,149,34,159),(25,158,35,148),(26,147,36,157),(27,156,37,146),(28,145,38,155),(29,154,39,144),(30,143,40,153),(41,111,51,101),(42,120,52,110),(43,109,53,119),(44,118,54,108),(45,107,55,117),(46,116,56,106),(47,105,57,115),(48,114,58,104),(49,103,59,113),(50,112,60,102),(81,122,91,132),(82,131,92,121),(83,140,93,130),(84,129,94,139),(85,138,95,128),(86,127,96,137),(87,136,97,126),(88,125,98,135),(89,134,99,124),(90,123,100,133)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 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 | 4 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.95D10 | C4×D20 | C42⋊2D5 | D10.12D4 | D10⋊D4 | C4.Dic10 | D5×C4⋊C4 | D20⋊8C4 | C4⋊D20 | C4×C5⋊D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.95D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 20 | 9 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 36 |
| 0 | 0 | 25 | 1 |
| 14 | 27 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 40 | 36 |
| 0 | 0 | 25 | 1 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,32,20,0,0,0,9],[24,40,0,0,1,17,0,0,0,0,9,0,0,0,0,9],[34,7,0,0,34,1,0,0,0,0,40,25,0,0,36,1],[14,11,0,0,27,27,0,0,0,0,40,25,0,0,36,1] >;
C42.95D10 in GAP, Magma, Sage, TeX
C_4^2._{95}D_{10} % in TeX
G:=Group("C4^2.95D10"); // GroupNames label
G:=SmallGroup(320,1202);
// by ID
G=gap.SmallGroup(320,1202);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations