metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.157D10, C10.312- 1+4, C10.1352+ 1+4, C20⋊Q8⋊38C2, C4⋊C4.114D10, C42.C2⋊13D5, D10⋊2Q8⋊39C2, D10⋊Q8⋊37C2, C20.6Q8⋊30C2, (C2×C20).190C23, (C2×C10).243C24, (C4×C20).224C22, C4.D20.14C2, (C2×D20).38C22, C2.60(D4⋊8D10), D10.13D4.4C2, C4⋊Dic5.245C22, C22.264(C23×D5), D10⋊C4.43C22, C5⋊4(C22.57C24), (C2×Dic10).43C22, (C2×Dic5).125C23, (C4×Dic5).156C22, C10.D4.86C22, (C22×D5).108C23, C2.61(D4.10D10), C2.32(Q8.10D10), C4⋊C4⋊D5⋊38C2, (C5×C42.C2)⋊16C2, (C2×C4×D5).142C22, (C5×C4⋊C4).198C22, (C2×C4).207(C22×D5), SmallGroup(320,1371)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.157D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 710 in 196 conjugacy classes, 91 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22.57C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, C20.6Q8, C4.D20, C20⋊Q8, D10.13D4, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42.C2, C42.157D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.57C24, C23×D5, Q8.10D10, D4⋊8D10, D4.10D10, C42.157D10
►On 160 points
Generators in S160
(1 143 122 35)(2 154 123 26)(3 145 124 37)(4 156 125 28)(5 147 126 39)(6 158 127 30)(7 149 128 21)(8 160 129 32)(9 151 130 23)(10 142 131 34)(11 153 132 25)(12 144 133 36)(13 155 134 27)(14 146 135 38)(15 157 136 29)(16 148 137 40)(17 159 138 31)(18 150 139 22)(19 141 140 33)(20 152 121 24)(41 90 115 73)(42 81 116 64)(43 92 117 75)(44 83 118 66)(45 94 119 77)(46 85 120 68)(47 96 101 79)(48 87 102 70)(49 98 103 61)(50 89 104 72)(51 100 105 63)(52 91 106 74)(53 82 107 65)(54 93 108 76)(55 84 109 67)(56 95 110 78)(57 86 111 69)(58 97 112 80)(59 88 113 71)(60 99 114 62)
(1 44 132 108)(2 109 133 45)(3 46 134 110)(4 111 135 47)(5 48 136 112)(6 113 137 49)(7 50 138 114)(8 115 139 51)(9 52 140 116)(10 117 121 53)(11 54 122 118)(12 119 123 55)(13 56 124 120)(14 101 125 57)(15 58 126 102)(16 103 127 59)(17 60 128 104)(18 105 129 41)(19 42 130 106)(20 107 131 43)(21 72 159 99)(22 100 160 73)(23 74 141 81)(24 82 142 75)(25 76 143 83)(26 84 144 77)(27 78 145 85)(28 86 146 79)(29 80 147 87)(30 88 148 61)(31 62 149 89)(32 90 150 63)(33 64 151 91)(34 92 152 65)(35 66 153 93)(36 94 154 67)(37 68 155 95)(38 96 156 69)(39 70 157 97)(40 98 158 71)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 156 31 146)(22 145 32 155)(23 154 33 144)(24 143 34 153)(25 152 35 142)(26 141 36 151)(27 150 37 160)(28 159 38 149)(29 148 39 158)(30 157 40 147)(41 120 51 110)(42 109 52 119)(43 118 53 108)(44 107 54 117)(45 116 55 106)(46 105 56 115)(47 114 57 104)(48 103 58 113)(49 112 59 102)(50 101 60 111)(61 80 71 70)(62 69 72 79)(63 78 73 68)(64 67 74 77)(65 76 75 66)(81 84 91 94)(82 93 92 83)(85 100 95 90)(86 89 96 99)(87 98 97 88)(121 122 131 132)(123 140 133 130)(124 129 134 139)(125 138 135 128)(126 127 136 137)
G:=sub<Sym(160)| (1,143,122,35)(2,154,123,26)(3,145,124,37)(4,156,125,28)(5,147,126,39)(6,158,127,30)(7,149,128,21)(8,160,129,32)(9,151,130,23)(10,142,131,34)(11,153,132,25)(12,144,133,36)(13,155,134,27)(14,146,135,38)(15,157,136,29)(16,148,137,40)(17,159,138,31)(18,150,139,22)(19,141,140,33)(20,152,121,24)(41,90,115,73)(42,81,116,64)(43,92,117,75)(44,83,118,66)(45,94,119,77)(46,85,120,68)(47,96,101,79)(48,87,102,70)(49,98,103,61)(50,89,104,72)(51,100,105,63)(52,91,106,74)(53,82,107,65)(54,93,108,76)(55,84,109,67)(56,95,110,78)(57,86,111,69)(58,97,112,80)(59,88,113,71)(60,99,114,62), (1,44,132,108)(2,109,133,45)(3,46,134,110)(4,111,135,47)(5,48,136,112)(6,113,137,49)(7,50,138,114)(8,115,139,51)(9,52,140,116)(10,117,121,53)(11,54,122,118)(12,119,123,55)(13,56,124,120)(14,101,125,57)(15,58,126,102)(16,103,127,59)(17,60,128,104)(18,105,129,41)(19,42,130,106)(20,107,131,43)(21,72,159,99)(22,100,160,73)(23,74,141,81)(24,82,142,75)(25,76,143,83)(26,84,144,77)(27,78,145,85)(28,86,146,79)(29,80,147,87)(30,88,148,61)(31,62,149,89)(32,90,150,63)(33,64,151,91)(34,92,152,65)(35,66,153,93)(36,94,154,67)(37,68,155,95)(38,96,156,69)(39,70,157,97)(40,98,158,71), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,156,31,146)(22,145,32,155)(23,154,33,144)(24,143,34,153)(25,152,35,142)(26,141,36,151)(27,150,37,160)(28,159,38,149)(29,148,39,158)(30,157,40,147)(41,120,51,110)(42,109,52,119)(43,118,53,108)(44,107,54,117)(45,116,55,106)(46,105,56,115)(47,114,57,104)(48,103,58,113)(49,112,59,102)(50,101,60,111)(61,80,71,70)(62,69,72,79)(63,78,73,68)(64,67,74,77)(65,76,75,66)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137)>;
G:=Group( (1,143,122,35)(2,154,123,26)(3,145,124,37)(4,156,125,28)(5,147,126,39)(6,158,127,30)(7,149,128,21)(8,160,129,32)(9,151,130,23)(10,142,131,34)(11,153,132,25)(12,144,133,36)(13,155,134,27)(14,146,135,38)(15,157,136,29)(16,148,137,40)(17,159,138,31)(18,150,139,22)(19,141,140,33)(20,152,121,24)(41,90,115,73)(42,81,116,64)(43,92,117,75)(44,83,118,66)(45,94,119,77)(46,85,120,68)(47,96,101,79)(48,87,102,70)(49,98,103,61)(50,89,104,72)(51,100,105,63)(52,91,106,74)(53,82,107,65)(54,93,108,76)(55,84,109,67)(56,95,110,78)(57,86,111,69)(58,97,112,80)(59,88,113,71)(60,99,114,62), (1,44,132,108)(2,109,133,45)(3,46,134,110)(4,111,135,47)(5,48,136,112)(6,113,137,49)(7,50,138,114)(8,115,139,51)(9,52,140,116)(10,117,121,53)(11,54,122,118)(12,119,123,55)(13,56,124,120)(14,101,125,57)(15,58,126,102)(16,103,127,59)(17,60,128,104)(18,105,129,41)(19,42,130,106)(20,107,131,43)(21,72,159,99)(22,100,160,73)(23,74,141,81)(24,82,142,75)(25,76,143,83)(26,84,144,77)(27,78,145,85)(28,86,146,79)(29,80,147,87)(30,88,148,61)(31,62,149,89)(32,90,150,63)(33,64,151,91)(34,92,152,65)(35,66,153,93)(36,94,154,67)(37,68,155,95)(38,96,156,69)(39,70,157,97)(40,98,158,71), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,156,31,146)(22,145,32,155)(23,154,33,144)(24,143,34,153)(25,152,35,142)(26,141,36,151)(27,150,37,160)(28,159,38,149)(29,148,39,158)(30,157,40,147)(41,120,51,110)(42,109,52,119)(43,118,53,108)(44,107,54,117)(45,116,55,106)(46,105,56,115)(47,114,57,104)(48,103,58,113)(49,112,59,102)(50,101,60,111)(61,80,71,70)(62,69,72,79)(63,78,73,68)(64,67,74,77)(65,76,75,66)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137) );
G=PermutationGroup([[(1,143,122,35),(2,154,123,26),(3,145,124,37),(4,156,125,28),(5,147,126,39),(6,158,127,30),(7,149,128,21),(8,160,129,32),(9,151,130,23),(10,142,131,34),(11,153,132,25),(12,144,133,36),(13,155,134,27),(14,146,135,38),(15,157,136,29),(16,148,137,40),(17,159,138,31),(18,150,139,22),(19,141,140,33),(20,152,121,24),(41,90,115,73),(42,81,116,64),(43,92,117,75),(44,83,118,66),(45,94,119,77),(46,85,120,68),(47,96,101,79),(48,87,102,70),(49,98,103,61),(50,89,104,72),(51,100,105,63),(52,91,106,74),(53,82,107,65),(54,93,108,76),(55,84,109,67),(56,95,110,78),(57,86,111,69),(58,97,112,80),(59,88,113,71),(60,99,114,62)], [(1,44,132,108),(2,109,133,45),(3,46,134,110),(4,111,135,47),(5,48,136,112),(6,113,137,49),(7,50,138,114),(8,115,139,51),(9,52,140,116),(10,117,121,53),(11,54,122,118),(12,119,123,55),(13,56,124,120),(14,101,125,57),(15,58,126,102),(16,103,127,59),(17,60,128,104),(18,105,129,41),(19,42,130,106),(20,107,131,43),(21,72,159,99),(22,100,160,73),(23,74,141,81),(24,82,142,75),(25,76,143,83),(26,84,144,77),(27,78,145,85),(28,86,146,79),(29,80,147,87),(30,88,148,61),(31,62,149,89),(32,90,150,63),(33,64,151,91),(34,92,152,65),(35,66,153,93),(36,94,154,67),(37,68,155,95),(38,96,156,69),(39,70,157,97),(40,98,158,71)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,156,31,146),(22,145,32,155),(23,154,33,144),(24,143,34,153),(25,152,35,142),(26,141,36,151),(27,150,37,160),(28,159,38,149),(29,148,39,158),(30,157,40,147),(41,120,51,110),(42,109,52,119),(43,118,53,108),(44,107,54,117),(45,116,55,106),(46,105,56,115),(47,114,57,104),(48,103,58,113),(49,112,59,102),(50,101,60,111),(61,80,71,70),(62,69,72,79),(63,78,73,68),(64,67,74,77),(65,76,75,66),(81,84,91,94),(82,93,92,83),(85,100,95,90),(86,89,96,99),(87,98,97,88),(121,122,131,132),(123,140,133,130),(124,129,134,139),(125,138,135,128),(126,127,136,137)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4G | 4H | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2+ 1+4 | 2- 1+4 | Q8.10D10 | D4⋊8D10 | D4.10D10 |
| kernel | C42.157D10 | C20.6Q8 | C4.D20 | C20⋊Q8 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C42 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 12 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of C42.157D10 ►in GL8(𝔽41)
| 39 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 8 | 32 | 0 |
| 0 | 0 | 0 | 0 | 31 | 5 | 0 | 32 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 20 | 17 | 40 |
| 0 | 0 | 0 | 0 | 39 | 32 | 1 | 24 |
| 31 | 31 | 6 | 35 | 0 | 0 | 0 | 0 |
| 10 | 12 | 6 | 11 | 0 | 0 | 0 | 0 |
| 6 | 35 | 10 | 10 | 0 | 0 | 0 | 0 |
| 6 | 11 | 31 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 1 | 27 | 27 |
| 0 | 0 | 0 | 0 | 15 | 16 | 14 | 2 |
| 0 | 0 | 0 | 0 | 36 | 35 | 11 | 33 |
| 0 | 0 | 0 | 0 | 37 | 33 | 39 | 22 |
| 31 | 31 | 6 | 35 | 0 | 0 | 0 | 0 |
| 12 | 10 | 30 | 35 | 0 | 0 | 0 | 0 |
| 35 | 6 | 31 | 31 | 0 | 0 | 0 | 0 |
| 11 | 6 | 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 1 | 27 | 27 |
| 0 | 0 | 0 | 0 | 0 | 18 | 2 | 14 |
| 0 | 0 | 0 | 0 | 2 | 4 | 33 | 11 |
| 0 | 0 | 0 | 0 | 29 | 26 | 22 | 39 |
G:=sub<GL(8,GF(41))| [39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,9,0,29,31,0,0,0,0,0,9,8,5,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,1,9,39,0,0,0,0,40,24,20,32,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24],[31,10,6,6,0,0,0,0,31,12,35,11,0,0,0,0,6,6,10,31,0,0,0,0,35,11,10,29,0,0,0,0,0,0,0,0,33,15,36,37,0,0,0,0,1,16,35,33,0,0,0,0,27,14,11,39,0,0,0,0,27,2,33,22],[31,12,35,11,0,0,0,0,31,10,6,6,0,0,0,0,6,30,31,12,0,0,0,0,35,35,31,10,0,0,0,0,0,0,0,0,33,0,2,29,0,0,0,0,1,18,4,26,0,0,0,0,27,2,33,22,0,0,0,0,27,14,11,39] >;
C42.157D10 in GAP, Magma, Sage, TeX
C_4^2._{157}D_{10} % in TeX
G:=Group("C4^2.157D10"); // GroupNames label
G:=SmallGroup(320,1371);
// by ID
G=gap.SmallGroup(320,1371);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,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