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), C4⋊D20⋊17C2, C4⋊2D20⋊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
Subgroups: 1230 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C4 [×8], C22, C22 [×16], C5, C2×C4, C2×C4 [×6], C2×C4 [×19], D4 [×20], Q8 [×4], C23 [×5], D5 [×6], C10, C10 [×2], C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×6], C20 [×4], D10 [×2], D10 [×14], C2×C10, C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], C4×D5 [×4], C4×D5 [×12], D20 [×20], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22×D5 [×4], C22.26C24, C4×Dic5, C4×Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5, C2×C4×D5 [×6], C2×D20 [×10], Q8⋊2D5 [×8], Q8×C10 [×2], D5×C42, C4⋊D20, D20⋊8C4 [×4], C4⋊2D20 [×4], C20.23D4 [×2], C5×C4⋊Q8, C2×Q8⋊2D5 [×2], C42.240D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], Q8⋊2D5 [×4], C23×D5, C2×D4×D5, C2×Q8⋊2D5 [×2], C42.240D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 58 133 92)(2 93 134 59)(3 60 135 94)(4 95 136 41)(5 42 137 96)(6 97 138 43)(7 44 139 98)(8 99 140 45)(9 46 121 100)(10 81 122 47)(11 48 123 82)(12 83 124 49)(13 50 125 84)(14 85 126 51)(15 52 127 86)(16 87 128 53)(17 54 129 88)(18 89 130 55)(19 56 131 90)(20 91 132 57)(21 106 76 156)(22 157 77 107)(23 108 78 158)(24 159 79 109)(25 110 80 160)(26 141 61 111)(27 112 62 142)(28 143 63 113)(29 114 64 144)(30 145 65 115)(31 116 66 146)(32 147 67 117)(33 118 68 148)(34 149 69 119)(35 120 70 150)(36 151 71 101)(37 102 72 152)(38 153 73 103)(39 104 74 154)(40 155 75 105)
(1 31 11 21)(2 22 12 32)(3 33 13 23)(4 24 14 34)(5 35 15 25)(6 26 16 36)(7 37 17 27)(8 28 18 38)(9 39 19 29)(10 30 20 40)(41 109 51 119)(42 120 52 110)(43 111 53 101)(44 102 54 112)(45 113 55 103)(46 104 56 114)(47 115 57 105)(48 106 58 116)(49 117 59 107)(50 108 60 118)(61 128 71 138)(62 139 72 129)(63 130 73 140)(64 121 74 131)(65 132 75 122)(66 123 76 133)(67 134 77 124)(68 125 78 135)(69 136 79 126)(70 127 80 137)(81 145 91 155)(82 156 92 146)(83 147 93 157)(84 158 94 148)(85 149 95 159)(86 160 96 150)(87 151 97 141)(88 142 98 152)(89 153 99 143)(90 144 100 154)
(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 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 84)(51 83)(52 82)(53 81)(54 100)(55 99)(56 98)(57 97)(58 96)(59 95)(60 94)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(101 155)(102 154)(103 153)(104 152)(105 151)(106 150)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(113 143)(114 142)(115 141)(116 160)(117 159)(118 158)(119 157)(120 156)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)
G:=sub<Sym(160)| (1,58,133,92)(2,93,134,59)(3,60,135,94)(4,95,136,41)(5,42,137,96)(6,97,138,43)(7,44,139,98)(8,99,140,45)(9,46,121,100)(10,81,122,47)(11,48,123,82)(12,83,124,49)(13,50,125,84)(14,85,126,51)(15,52,127,86)(16,87,128,53)(17,54,129,88)(18,89,130,55)(19,56,131,90)(20,91,132,57)(21,106,76,156)(22,157,77,107)(23,108,78,158)(24,159,79,109)(25,110,80,160)(26,141,61,111)(27,112,62,142)(28,143,63,113)(29,114,64,144)(30,145,65,115)(31,116,66,146)(32,147,67,117)(33,118,68,148)(34,149,69,119)(35,120,70,150)(36,151,71,101)(37,102,72,152)(38,153,73,103)(39,104,74,154)(40,155,75,105), (1,31,11,21)(2,22,12,32)(3,33,13,23)(4,24,14,34)(5,35,15,25)(6,26,16,36)(7,37,17,27)(8,28,18,38)(9,39,19,29)(10,30,20,40)(41,109,51,119)(42,120,52,110)(43,111,53,101)(44,102,54,112)(45,113,55,103)(46,104,56,114)(47,115,57,105)(48,106,58,116)(49,117,59,107)(50,108,60,118)(61,128,71,138)(62,139,72,129)(63,130,73,140)(64,121,74,131)(65,132,75,122)(66,123,76,133)(67,134,77,124)(68,125,78,135)(69,136,79,126)(70,127,80,137)(81,145,91,155)(82,156,92,146)(83,147,93,157)(84,158,94,148)(85,149,95,159)(86,160,96,150)(87,151,97,141)(88,142,98,152)(89,153,99,143)(90,144,100,154), (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)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,100)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)(116,160)(117,159)(118,158)(119,157)(120,156)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)>;
G:=Group( (1,58,133,92)(2,93,134,59)(3,60,135,94)(4,95,136,41)(5,42,137,96)(6,97,138,43)(7,44,139,98)(8,99,140,45)(9,46,121,100)(10,81,122,47)(11,48,123,82)(12,83,124,49)(13,50,125,84)(14,85,126,51)(15,52,127,86)(16,87,128,53)(17,54,129,88)(18,89,130,55)(19,56,131,90)(20,91,132,57)(21,106,76,156)(22,157,77,107)(23,108,78,158)(24,159,79,109)(25,110,80,160)(26,141,61,111)(27,112,62,142)(28,143,63,113)(29,114,64,144)(30,145,65,115)(31,116,66,146)(32,147,67,117)(33,118,68,148)(34,149,69,119)(35,120,70,150)(36,151,71,101)(37,102,72,152)(38,153,73,103)(39,104,74,154)(40,155,75,105), (1,31,11,21)(2,22,12,32)(3,33,13,23)(4,24,14,34)(5,35,15,25)(6,26,16,36)(7,37,17,27)(8,28,18,38)(9,39,19,29)(10,30,20,40)(41,109,51,119)(42,120,52,110)(43,111,53,101)(44,102,54,112)(45,113,55,103)(46,104,56,114)(47,115,57,105)(48,106,58,116)(49,117,59,107)(50,108,60,118)(61,128,71,138)(62,139,72,129)(63,130,73,140)(64,121,74,131)(65,132,75,122)(66,123,76,133)(67,134,77,124)(68,125,78,135)(69,136,79,126)(70,127,80,137)(81,145,91,155)(82,156,92,146)(83,147,93,157)(84,158,94,148)(85,149,95,159)(86,160,96,150)(87,151,97,141)(88,142,98,152)(89,153,99,143)(90,144,100,154), (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)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,100)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)(116,160)(117,159)(118,158)(119,157)(120,156)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136) );
G=PermutationGroup([(1,58,133,92),(2,93,134,59),(3,60,135,94),(4,95,136,41),(5,42,137,96),(6,97,138,43),(7,44,139,98),(8,99,140,45),(9,46,121,100),(10,81,122,47),(11,48,123,82),(12,83,124,49),(13,50,125,84),(14,85,126,51),(15,52,127,86),(16,87,128,53),(17,54,129,88),(18,89,130,55),(19,56,131,90),(20,91,132,57),(21,106,76,156),(22,157,77,107),(23,108,78,158),(24,159,79,109),(25,110,80,160),(26,141,61,111),(27,112,62,142),(28,143,63,113),(29,114,64,144),(30,145,65,115),(31,116,66,146),(32,147,67,117),(33,118,68,148),(34,149,69,119),(35,120,70,150),(36,151,71,101),(37,102,72,152),(38,153,73,103),(39,104,74,154),(40,155,75,105)], [(1,31,11,21),(2,22,12,32),(3,33,13,23),(4,24,14,34),(5,35,15,25),(6,26,16,36),(7,37,17,27),(8,28,18,38),(9,39,19,29),(10,30,20,40),(41,109,51,119),(42,120,52,110),(43,111,53,101),(44,102,54,112),(45,113,55,103),(46,104,56,114),(47,115,57,105),(48,106,58,116),(49,117,59,107),(50,108,60,118),(61,128,71,138),(62,139,72,129),(63,130,73,140),(64,121,74,131),(65,132,75,122),(66,123,76,133),(67,134,77,124),(68,125,78,135),(69,136,79,126),(70,127,80,137),(81,145,91,155),(82,156,92,146),(83,147,93,157),(84,158,94,148),(85,149,95,159),(86,160,96,150),(87,151,97,141),(88,142,98,152),(89,153,99,143),(90,144,100,154)], [(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),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,84),(51,83),(52,82),(53,81),(54,100),(55,99),(56,98),(57,97),(58,96),(59,95),(60,94),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(101,155),(102,154),(103,153),(104,152),(105,151),(106,150),(107,149),(108,148),(109,147),(110,146),(111,145),(112,144),(113,143),(114,142),(115,141),(116,160),(117,159),(118,158),(119,157),(120,156),(121,129),(122,128),(123,127),(124,126),(130,140),(131,139),(132,138),(133,137),(134,136)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D20 | D20⋊8C4 | C4⋊2D20 | 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 |
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
×
⋊
ℤ
𝔽
○
≀