metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.133D10, C10.132- 1+4, C10.1122+ 1+4, C20⋊Q8⋊17C2, (C4×Q8)⋊15D5, (C4×D20)⋊41C2, (Q8×C20)⋊17C2, C4⋊C4.300D10, D10⋊Q8⋊12C2, D10⋊3Q8⋊10C2, C4.49(C4○D20), C4⋊D20.11C2, C4.D20⋊29C2, C20.23D4⋊9C2, C42⋊2D5⋊12C2, C42⋊D5⋊18C2, (C2×Q8).181D10, D10.13D4⋊9C2, C20.120(C4○D4), (C2×C10).126C24, (C4×C20).178C22, (C2×C20).171C23, C2.24(D4⋊8D10), (C2×D20).271C22, D10⋊C4.7C22, C4⋊Dic5.369C22, (Q8×C10).226C22, (C2×Dic5).57C23, (C4×Dic5).94C22, (C22×D5).48C23, C22.147(C23×D5), C5⋊3(C22.36C24), (C2×Dic10).34C22, C10.D4.77C22, C2.14(Q8.10D10), C10.56(C2×C4○D4), C2.65(C2×C4○D20), (C2×C4×D5).85C22, (C5×C4⋊C4).354C22, (C2×C4).171(C22×D5), SmallGroup(320,1254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.133D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c9 >
Subgroups: 814 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22.36C24, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C42⋊D5, C4×D20, C4.D20, C42⋊2D5, C20⋊Q8, D10.13D4, C4⋊D20, D10⋊Q8, D10⋊3Q8, C20.23D4, Q8×C20, C42.133D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, C4○D20, C23×D5, C2×C4○D20, Q8.10D10, D4⋊8D10, C42.133D10
►On 160 points
Generators in S160
(1 119 11 109)(2 110 12 120)(3 101 13 111)(4 112 14 102)(5 103 15 113)(6 114 16 104)(7 105 17 115)(8 116 18 106)(9 107 19 117)(10 118 20 108)(21 41 31 51)(22 52 32 42)(23 43 33 53)(24 54 34 44)(25 45 35 55)(26 56 36 46)(27 47 37 57)(28 58 38 48)(29 49 39 59)(30 60 40 50)(61 83 71 93)(62 94 72 84)(63 85 73 95)(64 96 74 86)(65 87 75 97)(66 98 76 88)(67 89 77 99)(68 100 78 90)(69 91 79 81)(70 82 80 92)(121 151 131 141)(122 142 132 152)(123 153 133 143)(124 144 134 154)(125 155 135 145)(126 146 136 156)(127 157 137 147)(128 148 138 158)(129 159 139 149)(130 150 140 160)
(1 49 69 159)(2 50 70 160)(3 51 71 141)(4 52 72 142)(5 53 73 143)(6 54 74 144)(7 55 75 145)(8 56 76 146)(9 57 77 147)(10 58 78 148)(11 59 79 149)(12 60 80 150)(13 41 61 151)(14 42 62 152)(15 43 63 153)(16 44 64 154)(17 45 65 155)(18 46 66 156)(19 47 67 157)(20 48 68 158)(21 93 121 101)(22 94 122 102)(23 95 123 103)(24 96 124 104)(25 97 125 105)(26 98 126 106)(27 99 127 107)(28 100 128 108)(29 81 129 109)(30 82 130 110)(31 83 131 111)(32 84 132 112)(33 85 133 113)(34 86 134 114)(35 87 135 115)(36 88 136 116)(37 89 137 117)(38 90 138 118)(39 91 139 119)(40 92 140 120)
(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)
(1 123 79 33)(2 32 80 122)(3 121 61 31)(4 30 62 140)(5 139 63 29)(6 28 64 138)(7 137 65 27)(8 26 66 136)(9 135 67 25)(10 24 68 134)(11 133 69 23)(12 22 70 132)(13 131 71 21)(14 40 72 130)(15 129 73 39)(16 38 74 128)(17 127 75 37)(18 36 76 126)(19 125 77 35)(20 34 78 124)(41 101 141 83)(42 82 142 120)(43 119 143 81)(44 100 144 118)(45 117 145 99)(46 98 146 116)(47 115 147 97)(48 96 148 114)(49 113 149 95)(50 94 150 112)(51 111 151 93)(52 92 152 110)(53 109 153 91)(54 90 154 108)(55 107 155 89)(56 88 156 106)(57 105 157 87)(58 86 158 104)(59 103 159 85)(60 84 160 102)
G:=sub<Sym(160)| (1,119,11,109)(2,110,12,120)(3,101,13,111)(4,112,14,102)(5,103,15,113)(6,114,16,104)(7,105,17,115)(8,116,18,106)(9,107,19,117)(10,118,20,108)(21,41,31,51)(22,52,32,42)(23,43,33,53)(24,54,34,44)(25,45,35,55)(26,56,36,46)(27,47,37,57)(28,58,38,48)(29,49,39,59)(30,60,40,50)(61,83,71,93)(62,94,72,84)(63,85,73,95)(64,96,74,86)(65,87,75,97)(66,98,76,88)(67,89,77,99)(68,100,78,90)(69,91,79,81)(70,82,80,92)(121,151,131,141)(122,142,132,152)(123,153,133,143)(124,144,134,154)(125,155,135,145)(126,146,136,156)(127,157,137,147)(128,148,138,158)(129,159,139,149)(130,150,140,160), (1,49,69,159)(2,50,70,160)(3,51,71,141)(4,52,72,142)(5,53,73,143)(6,54,74,144)(7,55,75,145)(8,56,76,146)(9,57,77,147)(10,58,78,148)(11,59,79,149)(12,60,80,150)(13,41,61,151)(14,42,62,152)(15,43,63,153)(16,44,64,154)(17,45,65,155)(18,46,66,156)(19,47,67,157)(20,48,68,158)(21,93,121,101)(22,94,122,102)(23,95,123,103)(24,96,124,104)(25,97,125,105)(26,98,126,106)(27,99,127,107)(28,100,128,108)(29,81,129,109)(30,82,130,110)(31,83,131,111)(32,84,132,112)(33,85,133,113)(34,86,134,114)(35,87,135,115)(36,88,136,116)(37,89,137,117)(38,90,138,118)(39,91,139,119)(40,92,140,120), (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), (1,123,79,33)(2,32,80,122)(3,121,61,31)(4,30,62,140)(5,139,63,29)(6,28,64,138)(7,137,65,27)(8,26,66,136)(9,135,67,25)(10,24,68,134)(11,133,69,23)(12,22,70,132)(13,131,71,21)(14,40,72,130)(15,129,73,39)(16,38,74,128)(17,127,75,37)(18,36,76,126)(19,125,77,35)(20,34,78,124)(41,101,141,83)(42,82,142,120)(43,119,143,81)(44,100,144,118)(45,117,145,99)(46,98,146,116)(47,115,147,97)(48,96,148,114)(49,113,149,95)(50,94,150,112)(51,111,151,93)(52,92,152,110)(53,109,153,91)(54,90,154,108)(55,107,155,89)(56,88,156,106)(57,105,157,87)(58,86,158,104)(59,103,159,85)(60,84,160,102)>;
G:=Group( (1,119,11,109)(2,110,12,120)(3,101,13,111)(4,112,14,102)(5,103,15,113)(6,114,16,104)(7,105,17,115)(8,116,18,106)(9,107,19,117)(10,118,20,108)(21,41,31,51)(22,52,32,42)(23,43,33,53)(24,54,34,44)(25,45,35,55)(26,56,36,46)(27,47,37,57)(28,58,38,48)(29,49,39,59)(30,60,40,50)(61,83,71,93)(62,94,72,84)(63,85,73,95)(64,96,74,86)(65,87,75,97)(66,98,76,88)(67,89,77,99)(68,100,78,90)(69,91,79,81)(70,82,80,92)(121,151,131,141)(122,142,132,152)(123,153,133,143)(124,144,134,154)(125,155,135,145)(126,146,136,156)(127,157,137,147)(128,148,138,158)(129,159,139,149)(130,150,140,160), (1,49,69,159)(2,50,70,160)(3,51,71,141)(4,52,72,142)(5,53,73,143)(6,54,74,144)(7,55,75,145)(8,56,76,146)(9,57,77,147)(10,58,78,148)(11,59,79,149)(12,60,80,150)(13,41,61,151)(14,42,62,152)(15,43,63,153)(16,44,64,154)(17,45,65,155)(18,46,66,156)(19,47,67,157)(20,48,68,158)(21,93,121,101)(22,94,122,102)(23,95,123,103)(24,96,124,104)(25,97,125,105)(26,98,126,106)(27,99,127,107)(28,100,128,108)(29,81,129,109)(30,82,130,110)(31,83,131,111)(32,84,132,112)(33,85,133,113)(34,86,134,114)(35,87,135,115)(36,88,136,116)(37,89,137,117)(38,90,138,118)(39,91,139,119)(40,92,140,120), (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), (1,123,79,33)(2,32,80,122)(3,121,61,31)(4,30,62,140)(5,139,63,29)(6,28,64,138)(7,137,65,27)(8,26,66,136)(9,135,67,25)(10,24,68,134)(11,133,69,23)(12,22,70,132)(13,131,71,21)(14,40,72,130)(15,129,73,39)(16,38,74,128)(17,127,75,37)(18,36,76,126)(19,125,77,35)(20,34,78,124)(41,101,141,83)(42,82,142,120)(43,119,143,81)(44,100,144,118)(45,117,145,99)(46,98,146,116)(47,115,147,97)(48,96,148,114)(49,113,149,95)(50,94,150,112)(51,111,151,93)(52,92,152,110)(53,109,153,91)(54,90,154,108)(55,107,155,89)(56,88,156,106)(57,105,157,87)(58,86,158,104)(59,103,159,85)(60,84,160,102) );
G=PermutationGroup([[(1,119,11,109),(2,110,12,120),(3,101,13,111),(4,112,14,102),(5,103,15,113),(6,114,16,104),(7,105,17,115),(8,116,18,106),(9,107,19,117),(10,118,20,108),(21,41,31,51),(22,52,32,42),(23,43,33,53),(24,54,34,44),(25,45,35,55),(26,56,36,46),(27,47,37,57),(28,58,38,48),(29,49,39,59),(30,60,40,50),(61,83,71,93),(62,94,72,84),(63,85,73,95),(64,96,74,86),(65,87,75,97),(66,98,76,88),(67,89,77,99),(68,100,78,90),(69,91,79,81),(70,82,80,92),(121,151,131,141),(122,142,132,152),(123,153,133,143),(124,144,134,154),(125,155,135,145),(126,146,136,156),(127,157,137,147),(128,148,138,158),(129,159,139,149),(130,150,140,160)], [(1,49,69,159),(2,50,70,160),(3,51,71,141),(4,52,72,142),(5,53,73,143),(6,54,74,144),(7,55,75,145),(8,56,76,146),(9,57,77,147),(10,58,78,148),(11,59,79,149),(12,60,80,150),(13,41,61,151),(14,42,62,152),(15,43,63,153),(16,44,64,154),(17,45,65,155),(18,46,66,156),(19,47,67,157),(20,48,68,158),(21,93,121,101),(22,94,122,102),(23,95,123,103),(24,96,124,104),(25,97,125,105),(26,98,126,106),(27,99,127,107),(28,100,128,108),(29,81,129,109),(30,82,130,110),(31,83,131,111),(32,84,132,112),(33,85,133,113),(34,86,134,114),(35,87,135,115),(36,88,136,116),(37,89,137,117),(38,90,138,118),(39,91,139,119),(40,92,140,120)], [(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)], [(1,123,79,33),(2,32,80,122),(3,121,61,31),(4,30,62,140),(5,139,63,29),(6,28,64,138),(7,137,65,27),(8,26,66,136),(9,135,67,25),(10,24,68,134),(11,133,69,23),(12,22,70,132),(13,131,71,21),(14,40,72,130),(15,129,73,39),(16,38,74,128),(17,127,75,37),(18,36,76,126),(19,125,77,35),(20,34,78,124),(41,101,141,83),(42,82,142,120),(43,119,143,81),(44,100,144,118),(45,117,145,99),(46,98,146,116),(47,115,147,97),(48,96,148,114),(49,113,149,95),(50,94,150,112),(51,111,151,93),(52,92,152,110),(53,109,153,91),(54,90,154,108),(55,107,155,89),(56,88,156,106),(57,105,157,87),(58,86,158,104),(59,103,159,85),(60,84,160,102)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 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 | 20 | 20 | 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 | 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 | 2- 1+4 | Q8.10D10 | D4⋊8D10 |
| kernel | C42.133D10 | C42⋊D5 | C4×D20 | C4.D20 | C42⋊2D5 | C20⋊Q8 | D10.13D4 | C4⋊D20 | D10⋊Q8 | D10⋊3Q8 | C20.23D4 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C42.133D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 14 | 31 | 14 |
| 0 | 0 | 10 | 3 | 10 | 4 |
| 0 | 0 | 33 | 15 | 11 | 27 |
| 0 | 0 | 40 | 7 | 31 | 38 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 32 | 0 | 0 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 28 | 0 | 39 | 32 |
| 0 | 0 | 28 | 13 | 37 | 2 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 9 | 37 | 13 |
| 0 | 0 | 4 | 4 | 0 | 9 |
| 0 | 0 | 2 | 15 | 5 | 32 |
| 0 | 0 | 40 | 17 | 20 | 13 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 4 | 5 | 36 |
| 0 | 0 | 3 | 17 | 30 | 6 |
| 0 | 0 | 31 | 23 | 20 | 32 |
| 0 | 0 | 24 | 40 | 38 | 10 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,10,33,40,0,0,14,3,15,7,0,0,31,10,11,31,0,0,14,4,27,38],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,11,9,28,28,0,0,32,30,0,13,0,0,0,0,39,37,0,0,0,0,32,2],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,19,4,2,40,0,0,9,4,15,17,0,0,37,0,5,20,0,0,13,9,32,13],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,35,3,31,24,0,0,4,17,23,40,0,0,5,30,20,38,0,0,36,6,32,10] >;
C42.133D10 in GAP, Magma, Sage, TeX
C_4^2._{133}D_{10} % in TeX
G:=Group("C4^2.133D10"); // GroupNames label
G:=SmallGroup(320,1254);
// by ID
G=gap.SmallGroup(320,1254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^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