metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.148D10, C10.952- 1+4, C10.1302+ 1+4, (C4×D5)⋊1Q8, C20⋊Q8⋊35C2, C4.39(Q8×D5), D10.4(C2×Q8), C42.C2⋊4D5, C20.50(C2×Q8), C4⋊C4.111D10, C20⋊2Q8⋊32C2, Dic5.6(C2×Q8), (C2×C20).87C23, D10⋊Q8.2C2, C4.Dic10⋊33C2, C42⋊D5.6C2, C10.42(C22×Q8), (C2×C10).233C24, (C4×C20).193C22, D10⋊2Q8.12C2, C2.55(D4⋊8D10), Dic5.Q8⋊32C2, C4⋊Dic5.240C22, C22.254(C23×D5), D10⋊C4.39C22, C5⋊4(C23.41C23), (C4×Dic5).148C22, (C2×Dic5).121C23, (C2×Dic10).42C22, (C22×D5).230C23, C2.57(D4.10D10), C10.D4.122C22, C2.25(C2×Q8×D5), (D5×C4⋊C4).11C2, (C5×C42.C2)⋊6C2, C4⋊C4⋊7D5.12C2, (C2×C4×D5).133C22, (C5×C4⋊C4).188C22, (C2×C4).203(C22×D5), SmallGroup(320,1361)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.148D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
Subgroups: 686 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.41C23, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C20⋊2Q8, C42⋊D5, C20⋊Q8, C20⋊Q8, Dic5.Q8, C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, D10⋊Q8, D10⋊2Q8, C5×C42.C2, C42.148D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, 2- 1+4, C22×D5, C23.41C23, Q8×D5, C23×D5, C2×Q8×D5, D4⋊8D10, D4.10D10, C42.148D10
►On 160 points
Generators in S160
(1 77 111 141)(2 142 112 78)(3 79 113 143)(4 144 114 80)(5 61 115 145)(6 146 116 62)(7 63 117 147)(8 148 118 64)(9 65 119 149)(10 150 120 66)(11 67 101 151)(12 152 102 68)(13 69 103 153)(14 154 104 70)(15 71 105 155)(16 156 106 72)(17 73 107 157)(18 158 108 74)(19 75 109 159)(20 160 110 76)(21 89 47 126)(22 127 48 90)(23 91 49 128)(24 129 50 92)(25 93 51 130)(26 131 52 94)(27 95 53 132)(28 133 54 96)(29 97 55 134)(30 135 56 98)(31 99 57 136)(32 137 58 100)(33 81 59 138)(34 139 60 82)(35 83 41 140)(36 121 42 84)(37 85 43 122)(38 123 44 86)(39 87 45 124)(40 125 46 88)
(1 24 11 34)(2 41 12 51)(3 26 13 36)(4 43 14 53)(5 28 15 38)(6 45 16 55)(7 30 17 40)(8 47 18 57)(9 32 19 22)(10 49 20 59)(21 108 31 118)(23 110 33 120)(25 112 35 102)(27 114 37 104)(29 116 39 106)(42 113 52 103)(44 115 54 105)(46 117 56 107)(48 119 58 109)(50 101 60 111)(61 133 71 123)(62 87 72 97)(63 135 73 125)(64 89 74 99)(65 137 75 127)(66 91 76 81)(67 139 77 129)(68 93 78 83)(69 121 79 131)(70 95 80 85)(82 141 92 151)(84 143 94 153)(86 145 96 155)(88 147 98 157)(90 149 100 159)(122 154 132 144)(124 156 134 146)(126 158 136 148)(128 160 138 150)(130 142 140 152)
(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 52 31 42)(22 41 32 51)(23 50 33 60)(24 59 34 49)(25 48 35 58)(26 57 36 47)(27 46 37 56)(28 55 38 45)(29 44 39 54)(30 53 40 43)(61 62 71 72)(63 80 73 70)(64 69 74 79)(65 78 75 68)(66 67 76 77)(81 139 91 129)(82 128 92 138)(83 137 93 127)(84 126 94 136)(85 135 95 125)(86 124 96 134)(87 133 97 123)(88 122 98 132)(89 131 99 121)(90 140 100 130)(101 110 111 120)(102 119 112 109)(103 108 113 118)(104 117 114 107)(105 106 115 116)(141 150 151 160)(142 159 152 149)(143 148 153 158)(144 157 154 147)(145 146 155 156)
G:=sub<Sym(160)| (1,77,111,141)(2,142,112,78)(3,79,113,143)(4,144,114,80)(5,61,115,145)(6,146,116,62)(7,63,117,147)(8,148,118,64)(9,65,119,149)(10,150,120,66)(11,67,101,151)(12,152,102,68)(13,69,103,153)(14,154,104,70)(15,71,105,155)(16,156,106,72)(17,73,107,157)(18,158,108,74)(19,75,109,159)(20,160,110,76)(21,89,47,126)(22,127,48,90)(23,91,49,128)(24,129,50,92)(25,93,51,130)(26,131,52,94)(27,95,53,132)(28,133,54,96)(29,97,55,134)(30,135,56,98)(31,99,57,136)(32,137,58,100)(33,81,59,138)(34,139,60,82)(35,83,41,140)(36,121,42,84)(37,85,43,122)(38,123,44,86)(39,87,45,124)(40,125,46,88), (1,24,11,34)(2,41,12,51)(3,26,13,36)(4,43,14,53)(5,28,15,38)(6,45,16,55)(7,30,17,40)(8,47,18,57)(9,32,19,22)(10,49,20,59)(21,108,31,118)(23,110,33,120)(25,112,35,102)(27,114,37,104)(29,116,39,106)(42,113,52,103)(44,115,54,105)(46,117,56,107)(48,119,58,109)(50,101,60,111)(61,133,71,123)(62,87,72,97)(63,135,73,125)(64,89,74,99)(65,137,75,127)(66,91,76,81)(67,139,77,129)(68,93,78,83)(69,121,79,131)(70,95,80,85)(82,141,92,151)(84,143,94,153)(86,145,96,155)(88,147,98,157)(90,149,100,159)(122,154,132,144)(124,156,134,146)(126,158,136,148)(128,160,138,150)(130,142,140,152), (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)(21,52,31,42)(22,41,32,51)(23,50,33,60)(24,59,34,49)(25,48,35,58)(26,57,36,47)(27,46,37,56)(28,55,38,45)(29,44,39,54)(30,53,40,43)(61,62,71,72)(63,80,73,70)(64,69,74,79)(65,78,75,68)(66,67,76,77)(81,139,91,129)(82,128,92,138)(83,137,93,127)(84,126,94,136)(85,135,95,125)(86,124,96,134)(87,133,97,123)(88,122,98,132)(89,131,99,121)(90,140,100,130)(101,110,111,120)(102,119,112,109)(103,108,113,118)(104,117,114,107)(105,106,115,116)(141,150,151,160)(142,159,152,149)(143,148,153,158)(144,157,154,147)(145,146,155,156)>;
G:=Group( (1,77,111,141)(2,142,112,78)(3,79,113,143)(4,144,114,80)(5,61,115,145)(6,146,116,62)(7,63,117,147)(8,148,118,64)(9,65,119,149)(10,150,120,66)(11,67,101,151)(12,152,102,68)(13,69,103,153)(14,154,104,70)(15,71,105,155)(16,156,106,72)(17,73,107,157)(18,158,108,74)(19,75,109,159)(20,160,110,76)(21,89,47,126)(22,127,48,90)(23,91,49,128)(24,129,50,92)(25,93,51,130)(26,131,52,94)(27,95,53,132)(28,133,54,96)(29,97,55,134)(30,135,56,98)(31,99,57,136)(32,137,58,100)(33,81,59,138)(34,139,60,82)(35,83,41,140)(36,121,42,84)(37,85,43,122)(38,123,44,86)(39,87,45,124)(40,125,46,88), (1,24,11,34)(2,41,12,51)(3,26,13,36)(4,43,14,53)(5,28,15,38)(6,45,16,55)(7,30,17,40)(8,47,18,57)(9,32,19,22)(10,49,20,59)(21,108,31,118)(23,110,33,120)(25,112,35,102)(27,114,37,104)(29,116,39,106)(42,113,52,103)(44,115,54,105)(46,117,56,107)(48,119,58,109)(50,101,60,111)(61,133,71,123)(62,87,72,97)(63,135,73,125)(64,89,74,99)(65,137,75,127)(66,91,76,81)(67,139,77,129)(68,93,78,83)(69,121,79,131)(70,95,80,85)(82,141,92,151)(84,143,94,153)(86,145,96,155)(88,147,98,157)(90,149,100,159)(122,154,132,144)(124,156,134,146)(126,158,136,148)(128,160,138,150)(130,142,140,152), (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)(21,52,31,42)(22,41,32,51)(23,50,33,60)(24,59,34,49)(25,48,35,58)(26,57,36,47)(27,46,37,56)(28,55,38,45)(29,44,39,54)(30,53,40,43)(61,62,71,72)(63,80,73,70)(64,69,74,79)(65,78,75,68)(66,67,76,77)(81,139,91,129)(82,128,92,138)(83,137,93,127)(84,126,94,136)(85,135,95,125)(86,124,96,134)(87,133,97,123)(88,122,98,132)(89,131,99,121)(90,140,100,130)(101,110,111,120)(102,119,112,109)(103,108,113,118)(104,117,114,107)(105,106,115,116)(141,150,151,160)(142,159,152,149)(143,148,153,158)(144,157,154,147)(145,146,155,156) );
G=PermutationGroup([[(1,77,111,141),(2,142,112,78),(3,79,113,143),(4,144,114,80),(5,61,115,145),(6,146,116,62),(7,63,117,147),(8,148,118,64),(9,65,119,149),(10,150,120,66),(11,67,101,151),(12,152,102,68),(13,69,103,153),(14,154,104,70),(15,71,105,155),(16,156,106,72),(17,73,107,157),(18,158,108,74),(19,75,109,159),(20,160,110,76),(21,89,47,126),(22,127,48,90),(23,91,49,128),(24,129,50,92),(25,93,51,130),(26,131,52,94),(27,95,53,132),(28,133,54,96),(29,97,55,134),(30,135,56,98),(31,99,57,136),(32,137,58,100),(33,81,59,138),(34,139,60,82),(35,83,41,140),(36,121,42,84),(37,85,43,122),(38,123,44,86),(39,87,45,124),(40,125,46,88)], [(1,24,11,34),(2,41,12,51),(3,26,13,36),(4,43,14,53),(5,28,15,38),(6,45,16,55),(7,30,17,40),(8,47,18,57),(9,32,19,22),(10,49,20,59),(21,108,31,118),(23,110,33,120),(25,112,35,102),(27,114,37,104),(29,116,39,106),(42,113,52,103),(44,115,54,105),(46,117,56,107),(48,119,58,109),(50,101,60,111),(61,133,71,123),(62,87,72,97),(63,135,73,125),(64,89,74,99),(65,137,75,127),(66,91,76,81),(67,139,77,129),(68,93,78,83),(69,121,79,131),(70,95,80,85),(82,141,92,151),(84,143,94,153),(86,145,96,155),(88,147,98,157),(90,149,100,159),(122,154,132,144),(124,156,134,146),(126,158,136,148),(128,160,138,150),(130,142,140,152)], [(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),(21,52,31,42),(22,41,32,51),(23,50,33,60),(24,59,34,49),(25,48,35,58),(26,57,36,47),(27,46,37,56),(28,55,38,45),(29,44,39,54),(30,53,40,43),(61,62,71,72),(63,80,73,70),(64,69,74,79),(65,78,75,68),(66,67,76,77),(81,139,91,129),(82,128,92,138),(83,137,93,127),(84,126,94,136),(85,135,95,125),(86,124,96,134),(87,133,97,123),(88,122,98,132),(89,131,99,121),(90,140,100,130),(101,110,111,120),(102,119,112,109),(103,108,113,118),(104,117,114,107),(105,106,115,116),(141,150,151,160),(142,159,152,149),(143,148,153,158),(144,157,154,147),(145,146,155,156)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 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 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | 2+ 1+4 | 2- 1+4 | Q8×D5 | D4⋊8D10 | D4.10D10 |
| kernel | C42.148D10 | C20⋊2Q8 | C42⋊D5 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10⋊Q8 | D10⋊2Q8 | C5×C42.C2 | C4×D5 | C42.C2 | C42 | C4⋊C4 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 12 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.148D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 7 | 9 | 37 |
| 0 | 0 | 0 | 34 | 32 | 9 |
| 0 | 0 | 26 | 7 | 22 | 0 |
| 0 | 0 | 26 | 6 | 15 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 28 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 0 | 0 | 33 | 39 | 28 |
| 0 | 0 | 33 | 34 | 13 | 2 |
| 6 | 39 | 0 | 0 | 0 | 0 |
| 39 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 17 | 13 |
| 0 | 0 | 1 | 0 | 8 | 17 |
| 0 | 0 | 21 | 10 | 1 | 1 |
| 0 | 0 | 34 | 19 | 13 | 6 |
| 35 | 2 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 13 | 17 |
| 0 | 0 | 1 | 34 | 17 | 8 |
| 0 | 0 | 21 | 10 | 1 | 1 |
| 0 | 0 | 34 | 39 | 6 | 13 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,0,26,26,0,0,7,34,7,6,0,0,9,32,22,15,0,0,37,9,0,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,13,0,33,0,0,28,39,33,34,0,0,0,0,39,13,0,0,0,0,28,2],[6,39,0,0,0,0,39,35,0,0,0,0,0,0,34,1,21,34,0,0,7,0,10,19,0,0,17,8,1,13,0,0,13,17,1,6],[35,2,0,0,0,0,2,6,0,0,0,0,0,0,34,1,21,34,0,0,1,34,10,39,0,0,13,17,1,6,0,0,17,8,1,13] >;
C42.148D10 in GAP, Magma, Sage, TeX
C_4^2._{148}D_{10} % in TeX
G:=Group("C4^2.148D10"); // GroupNames label
G:=SmallGroup(320,1361);
// by ID
G=gap.SmallGroup(320,1361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,192,12550]);
// 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^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations