metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.240D10, (C4×D5)⋊9D4, C4⋊Q8⋊19D5, C4.38(D4×D5), C20⋊6(C4○D4), C20.70(C2×D4), C20⋊4D4⋊17C2, C4⋊D20⋊39C2, C4⋊C4.217D10, C4⋊1(Q8⋊2D5), D10.20(C2×D4), (D5×C42)⋊14C2, D20⋊8C4⋊43C2, (C2×Q8).144D10, C10.99(C22×D4), C20.23D4⋊26C2, (C2×C10).269C24, (C2×C20).102C23, (C4×C20).210C22, Dic5.122(C2×D4), (C2×D20).178C22, C5⋊6(C22.26C24), (Q8×C10).136C22, C22.290(C23×D5), D10⋊C4.50C22, (C2×Dic5).282C23, (C4×Dic5).289C22, (C22×D5).119C23, C2.72(C2×D4×D5), (C5×C4⋊Q8)⋊11C2, (C2×Q8⋊2D5)⋊12C2, C10.120(C2×C4○D4), C2.27(C2×Q8⋊2D5), (C2×C4×D5).152C22, (C5×C4⋊C4).212C22, (C2×C4).599(C22×D5), SmallGroup(320,1397)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.240D10
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c9 >
Subgroups: 1230 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22.26C24, C4×Dic5, C4×Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, Q8⋊2D5, Q8×C10, D5×C42, C20⋊4D4, D20⋊8C4, C4⋊D20, C20.23D4, C5×C4⋊Q8, C2×Q8⋊2D5, C42.240D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C22×D5, C22.26C24, D4×D5, Q8⋊2D5, C23×D5, C2×D4×D5, C2×Q8⋊2D5, C42.240D10
►On 160 points
Generators in S160
(1 137 110 74)(2 75 111 138)(3 139 112 76)(4 77 113 140)(5 121 114 78)(6 79 115 122)(7 123 116 80)(8 61 117 124)(9 125 118 62)(10 63 119 126)(11 127 120 64)(12 65 101 128)(13 129 102 66)(14 67 103 130)(15 131 104 68)(16 69 105 132)(17 133 106 70)(18 71 107 134)(19 135 108 72)(20 73 109 136)(21 156 93 41)(22 42 94 157)(23 158 95 43)(24 44 96 159)(25 160 97 45)(26 46 98 141)(27 142 99 47)(28 48 100 143)(29 144 81 49)(30 50 82 145)(31 146 83 51)(32 52 84 147)(33 148 85 53)(34 54 86 149)(35 150 87 55)(36 56 88 151)(37 152 89 57)(38 58 90 153)(39 154 91 59)(40 60 92 155)
(1 88 11 98)(2 99 12 89)(3 90 13 100)(4 81 14 91)(5 92 15 82)(6 83 16 93)(7 94 17 84)(8 85 18 95)(9 96 19 86)(10 87 20 97)(21 115 31 105)(22 106 32 116)(23 117 33 107)(24 108 34 118)(25 119 35 109)(26 110 36 120)(27 101 37 111)(28 112 38 102)(29 103 39 113)(30 114 40 104)(41 79 51 69)(42 70 52 80)(43 61 53 71)(44 72 54 62)(45 63 55 73)(46 74 56 64)(47 65 57 75)(48 76 58 66)(49 67 59 77)(50 78 60 68)(121 155 131 145)(122 146 132 156)(123 157 133 147)(124 148 134 158)(125 159 135 149)(126 150 136 160)(127 141 137 151)(128 152 138 142)(129 143 139 153)(130 154 140 144)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 160)(42 159)(43 158)(44 157)(45 156)(46 155)(47 154)(48 153)(49 152)(50 151)(51 150)(52 149)(53 148)(54 147)(55 146)(56 145)(57 144)(58 143)(59 142)(60 141)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 140)(76 139)(77 138)(78 137)(79 136)(80 135)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)
G:=sub<Sym(160)| (1,137,110,74)(2,75,111,138)(3,139,112,76)(4,77,113,140)(5,121,114,78)(6,79,115,122)(7,123,116,80)(8,61,117,124)(9,125,118,62)(10,63,119,126)(11,127,120,64)(12,65,101,128)(13,129,102,66)(14,67,103,130)(15,131,104,68)(16,69,105,132)(17,133,106,70)(18,71,107,134)(19,135,108,72)(20,73,109,136)(21,156,93,41)(22,42,94,157)(23,158,95,43)(24,44,96,159)(25,160,97,45)(26,46,98,141)(27,142,99,47)(28,48,100,143)(29,144,81,49)(30,50,82,145)(31,146,83,51)(32,52,84,147)(33,148,85,53)(34,54,86,149)(35,150,87,55)(36,56,88,151)(37,152,89,57)(38,58,90,153)(39,154,91,59)(40,60,92,155), (1,88,11,98)(2,99,12,89)(3,90,13,100)(4,81,14,91)(5,92,15,82)(6,83,16,93)(7,94,17,84)(8,85,18,95)(9,96,19,86)(10,87,20,97)(21,115,31,105)(22,106,32,116)(23,117,33,107)(24,108,34,118)(25,119,35,109)(26,110,36,120)(27,101,37,111)(28,112,38,102)(29,103,39,113)(30,114,40,104)(41,79,51,69)(42,70,52,80)(43,61,53,71)(44,72,54,62)(45,63,55,73)(46,74,56,64)(47,65,57,75)(48,76,58,66)(49,67,59,77)(50,78,60,68)(121,155,131,145)(122,146,132,156)(123,157,133,147)(124,148,134,158)(125,159,135,149)(126,150,136,160)(127,141,137,151)(128,152,138,142)(129,143,139,153)(130,154,140,144), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,89)(82,88)(83,87)(84,86)(90,100)(91,99)(92,98)(93,97)(94,96)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)>;
G:=Group( (1,137,110,74)(2,75,111,138)(3,139,112,76)(4,77,113,140)(5,121,114,78)(6,79,115,122)(7,123,116,80)(8,61,117,124)(9,125,118,62)(10,63,119,126)(11,127,120,64)(12,65,101,128)(13,129,102,66)(14,67,103,130)(15,131,104,68)(16,69,105,132)(17,133,106,70)(18,71,107,134)(19,135,108,72)(20,73,109,136)(21,156,93,41)(22,42,94,157)(23,158,95,43)(24,44,96,159)(25,160,97,45)(26,46,98,141)(27,142,99,47)(28,48,100,143)(29,144,81,49)(30,50,82,145)(31,146,83,51)(32,52,84,147)(33,148,85,53)(34,54,86,149)(35,150,87,55)(36,56,88,151)(37,152,89,57)(38,58,90,153)(39,154,91,59)(40,60,92,155), (1,88,11,98)(2,99,12,89)(3,90,13,100)(4,81,14,91)(5,92,15,82)(6,83,16,93)(7,94,17,84)(8,85,18,95)(9,96,19,86)(10,87,20,97)(21,115,31,105)(22,106,32,116)(23,117,33,107)(24,108,34,118)(25,119,35,109)(26,110,36,120)(27,101,37,111)(28,112,38,102)(29,103,39,113)(30,114,40,104)(41,79,51,69)(42,70,52,80)(43,61,53,71)(44,72,54,62)(45,63,55,73)(46,74,56,64)(47,65,57,75)(48,76,58,66)(49,67,59,77)(50,78,60,68)(121,155,131,145)(122,146,132,156)(123,157,133,147)(124,148,134,158)(125,159,135,149)(126,150,136,160)(127,141,137,151)(128,152,138,142)(129,143,139,153)(130,154,140,144), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,89)(82,88)(83,87)(84,86)(90,100)(91,99)(92,98)(93,97)(94,96)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113) );
G=PermutationGroup([[(1,137,110,74),(2,75,111,138),(3,139,112,76),(4,77,113,140),(5,121,114,78),(6,79,115,122),(7,123,116,80),(8,61,117,124),(9,125,118,62),(10,63,119,126),(11,127,120,64),(12,65,101,128),(13,129,102,66),(14,67,103,130),(15,131,104,68),(16,69,105,132),(17,133,106,70),(18,71,107,134),(19,135,108,72),(20,73,109,136),(21,156,93,41),(22,42,94,157),(23,158,95,43),(24,44,96,159),(25,160,97,45),(26,46,98,141),(27,142,99,47),(28,48,100,143),(29,144,81,49),(30,50,82,145),(31,146,83,51),(32,52,84,147),(33,148,85,53),(34,54,86,149),(35,150,87,55),(36,56,88,151),(37,152,89,57),(38,58,90,153),(39,154,91,59),(40,60,92,155)], [(1,88,11,98),(2,99,12,89),(3,90,13,100),(4,81,14,91),(5,92,15,82),(6,83,16,93),(7,94,17,84),(8,85,18,95),(9,96,19,86),(10,87,20,97),(21,115,31,105),(22,106,32,116),(23,117,33,107),(24,108,34,118),(25,119,35,109),(26,110,36,120),(27,101,37,111),(28,112,38,102),(29,103,39,113),(30,114,40,104),(41,79,51,69),(42,70,52,80),(43,61,53,71),(44,72,54,62),(45,63,55,73),(46,74,56,64),(47,65,57,75),(48,76,58,66),(49,67,59,77),(50,78,60,68),(121,155,131,145),(122,146,132,156),(123,157,133,147),(124,148,134,158),(125,159,135,149),(126,150,136,160),(127,141,137,151),(128,152,138,142),(129,143,139,153),(130,154,140,144)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,160),(42,159),(43,158),(44,157),(45,156),(46,155),(47,154),(48,153),(49,152),(50,151),(51,150),(52,149),(53,148),(54,147),(55,146),(56,145),(57,144),(58,143),(59,142),(60,141),(61,134),(62,133),(63,132),(64,131),(65,130),(66,129),(67,128),(68,127),(69,126),(70,125),(71,124),(72,123),(73,122),(74,121),(75,140),(76,139),(77,138),(78,137),(79,136),(80,135),(81,89),(82,88),(83,87),(84,86),(90,100),(91,99),(92,98),(93,97),(94,96),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D4×D5 | Q8⋊2D5 |
| kernel | C42.240D10 | D5×C42 | C20⋊4D4 | D20⋊8C4 | C4⋊D20 | C20.23D4 | C5×C4⋊Q8 | C2×Q8⋊2D5 | C4×D5 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 2 | 8 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C42.240D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 18 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,18,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,23,32,0,0,0,0,0,0,9,0,0,0,0,0,18,32],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,0,1] >;
C42.240D10 in GAP, Magma, Sage, TeX
C_4^2._{240}D_{10} % in TeX
G:=Group("C4^2.240D10"); // GroupNames label
G:=SmallGroup(320,1397);
// by ID
G=gap.SmallGroup(320,1397);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,185,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^9>;
// generators/relations