metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.30D10, C4⋊C8⋊15D5, C4.56(Q8×D5), C20⋊3C8⋊15C2, C4.209(D4×D5), (C4×D5).19Q8, (C4×D5).105D4, C20.368(C2×D4), (C2×C8).184D10, C20.114(C2×Q8), D10.13(C4⋊C4), C10.37(C8○D4), (C4×C20).65C22, C20.8Q8⋊24C2, C42⋊D5.1C2, D10⋊C4.24C4, Dic5.14(C4⋊C4), (C2×C20).836C23, (C2×C40).215C22, C10.D4.24C4, C5⋊3(C42.6C22), C2.14(D20.2C4), C2.15(D20.3C4), (C4×Dic5).205C22, (C5×C4⋊C8)⋊20C2, C2.10(D5×C4⋊C4), (D5×C2×C8).18C2, C10.32(C2×C4⋊C4), (C2×C4).36(C4×D5), C22.114(C2×C4×D5), (C2×C20).217(C2×C4), (C2×C8⋊D5).13C2, (C2×C4×D5).351C22, (C2×Dic5).99(C2×C4), (C22×D5).76(C2×C4), (C2×C4).778(C22×D5), (C2×C10).192(C22×C4), (C2×C5⋊2C8).316C22, SmallGroup(320,466)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.30D10
G = < a,b,c,d | a4=b4=1, c10=d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c9 >
Subgroups: 350 in 114 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4⋊C8, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.6C22, C8×D5, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C20⋊3C8, C20.8Q8, C5×C4⋊C8, C42⋊D5, D5×C2×C8, C2×C8⋊D5, C42.30D10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C8○D4, C4×D5, C22×D5, C42.6C22, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D20.3C4, D20.2C4, C42.30D10
►On 160 points
Generators in S160
(1 115 148 50)(2 51 149 116)(3 117 150 52)(4 53 151 118)(5 119 152 54)(6 55 153 120)(7 81 154 56)(8 57 155 82)(9 83 156 58)(10 59 157 84)(11 85 158 60)(12 61 159 86)(13 87 160 62)(14 63 121 88)(15 89 122 64)(16 65 123 90)(17 91 124 66)(18 67 125 92)(19 93 126 68)(20 69 127 94)(21 95 128 70)(22 71 129 96)(23 97 130 72)(24 73 131 98)(25 99 132 74)(26 75 133 100)(27 101 134 76)(28 77 135 102)(29 103 136 78)(30 79 137 104)(31 105 138 80)(32 41 139 106)(33 107 140 42)(34 43 141 108)(35 109 142 44)(36 45 143 110)(37 111 144 46)(38 47 145 112)(39 113 146 48)(40 49 147 114)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 11 20 21 30 31 40)(2 19 12 29 22 39 32 9)(3 28 13 38 23 8 33 18)(4 37 14 7 24 17 34 27)(5 6 15 16 25 26 35 36)(41 78 51 48 61 58 71 68)(42 47 52 57 62 67 72 77)(43 56 53 66 63 76 73 46)(44 65 54 75 64 45 74 55)(49 70 59 80 69 50 79 60)(81 118 91 88 101 98 111 108)(82 87 92 97 102 107 112 117)(83 96 93 106 103 116 113 86)(84 105 94 115 104 85 114 95)(89 110 99 120 109 90 119 100)(121 154 131 124 141 134 151 144)(122 123 132 133 142 143 152 153)(125 150 135 160 145 130 155 140)(126 159 136 129 146 139 156 149)(127 128 137 138 147 148 157 158)
G:=sub<Sym(160)| (1,115,148,50)(2,51,149,116)(3,117,150,52)(4,53,151,118)(5,119,152,54)(6,55,153,120)(7,81,154,56)(8,57,155,82)(9,83,156,58)(10,59,157,84)(11,85,158,60)(12,61,159,86)(13,87,160,62)(14,63,121,88)(15,89,122,64)(16,65,123,90)(17,91,124,66)(18,67,125,92)(19,93,126,68)(20,69,127,94)(21,95,128,70)(22,71,129,96)(23,97,130,72)(24,73,131,98)(25,99,132,74)(26,75,133,100)(27,101,134,76)(28,77,135,102)(29,103,136,78)(30,79,137,104)(31,105,138,80)(32,41,139,106)(33,107,140,42)(34,43,141,108)(35,109,142,44)(36,45,143,110)(37,111,144,46)(38,47,145,112)(39,113,146,48)(40,49,147,114), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,11,20,21,30,31,40)(2,19,12,29,22,39,32,9)(3,28,13,38,23,8,33,18)(4,37,14,7,24,17,34,27)(5,6,15,16,25,26,35,36)(41,78,51,48,61,58,71,68)(42,47,52,57,62,67,72,77)(43,56,53,66,63,76,73,46)(44,65,54,75,64,45,74,55)(49,70,59,80,69,50,79,60)(81,118,91,88,101,98,111,108)(82,87,92,97,102,107,112,117)(83,96,93,106,103,116,113,86)(84,105,94,115,104,85,114,95)(89,110,99,120,109,90,119,100)(121,154,131,124,141,134,151,144)(122,123,132,133,142,143,152,153)(125,150,135,160,145,130,155,140)(126,159,136,129,146,139,156,149)(127,128,137,138,147,148,157,158)>;
G:=Group( (1,115,148,50)(2,51,149,116)(3,117,150,52)(4,53,151,118)(5,119,152,54)(6,55,153,120)(7,81,154,56)(8,57,155,82)(9,83,156,58)(10,59,157,84)(11,85,158,60)(12,61,159,86)(13,87,160,62)(14,63,121,88)(15,89,122,64)(16,65,123,90)(17,91,124,66)(18,67,125,92)(19,93,126,68)(20,69,127,94)(21,95,128,70)(22,71,129,96)(23,97,130,72)(24,73,131,98)(25,99,132,74)(26,75,133,100)(27,101,134,76)(28,77,135,102)(29,103,136,78)(30,79,137,104)(31,105,138,80)(32,41,139,106)(33,107,140,42)(34,43,141,108)(35,109,142,44)(36,45,143,110)(37,111,144,46)(38,47,145,112)(39,113,146,48)(40,49,147,114), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,11,20,21,30,31,40)(2,19,12,29,22,39,32,9)(3,28,13,38,23,8,33,18)(4,37,14,7,24,17,34,27)(5,6,15,16,25,26,35,36)(41,78,51,48,61,58,71,68)(42,47,52,57,62,67,72,77)(43,56,53,66,63,76,73,46)(44,65,54,75,64,45,74,55)(49,70,59,80,69,50,79,60)(81,118,91,88,101,98,111,108)(82,87,92,97,102,107,112,117)(83,96,93,106,103,116,113,86)(84,105,94,115,104,85,114,95)(89,110,99,120,109,90,119,100)(121,154,131,124,141,134,151,144)(122,123,132,133,142,143,152,153)(125,150,135,160,145,130,155,140)(126,159,136,129,146,139,156,149)(127,128,137,138,147,148,157,158) );
G=PermutationGroup([[(1,115,148,50),(2,51,149,116),(3,117,150,52),(4,53,151,118),(5,119,152,54),(6,55,153,120),(7,81,154,56),(8,57,155,82),(9,83,156,58),(10,59,157,84),(11,85,158,60),(12,61,159,86),(13,87,160,62),(14,63,121,88),(15,89,122,64),(16,65,123,90),(17,91,124,66),(18,67,125,92),(19,93,126,68),(20,69,127,94),(21,95,128,70),(22,71,129,96),(23,97,130,72),(24,73,131,98),(25,99,132,74),(26,75,133,100),(27,101,134,76),(28,77,135,102),(29,103,136,78),(30,79,137,104),(31,105,138,80),(32,41,139,106),(33,107,140,42),(34,43,141,108),(35,109,142,44),(36,45,143,110),(37,111,144,46),(38,47,145,112),(39,113,146,48),(40,49,147,114)], [(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160)], [(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)], [(1,10,11,20,21,30,31,40),(2,19,12,29,22,39,32,9),(3,28,13,38,23,8,33,18),(4,37,14,7,24,17,34,27),(5,6,15,16,25,26,35,36),(41,78,51,48,61,58,71,68),(42,47,52,57,62,67,72,77),(43,56,53,66,63,76,73,46),(44,65,54,75,64,45,74,55),(49,70,59,80,69,50,79,60),(81,118,91,88,101,98,111,108),(82,87,92,97,102,107,112,117),(83,96,93,106,103,116,113,86),(84,105,94,115,104,85,114,95),(89,110,99,120,109,90,119,100),(121,154,131,124,141,134,151,144),(122,123,132,133,142,143,152,153),(125,150,135,160,145,130,155,140),(126,159,136,129,146,139,156,149),(127,128,137,138,147,148,157,158)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 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 | C4 | C4 | D4 | Q8 | D5 | D10 | D10 | C8○D4 | C4×D5 | D20.3C4 | D4×D5 | Q8×D5 | D20.2C4 |
| kernel | C42.30D10 | C20⋊3C8 | C20.8Q8 | C5×C4⋊C8 | C42⋊D5 | D5×C2×C8 | C2×C8⋊D5 | C10.D4 | D10⋊C4 | C4×D5 | C4×D5 | C4⋊C8 | C42 | C2×C8 | C10 | C2×C4 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C42.30D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 8 | 0 | 0 |
| 0 | 0 | 40 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 23 |
| 0 | 0 | 0 | 0 | 37 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 10 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 0 | 0 | 38 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,40,0,0,0,0,8,38,0,0,0,0,0,0,38,37,0,0,0,0,23,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,38,1,0,0,0,0,0,3],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,14,10,0,0,0,0,0,27,0,0,0,0,0,0,38,0,0,0,0,0,0,38] >;
C42.30D10 in GAP, Magma, Sage, TeX
C_4^2._{30}D_{10} % in TeX
G:=Group("C4^2.30D10"); // GroupNames label
G:=SmallGroup(320,466);
// by ID
G=gap.SmallGroup(320,466);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations