metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊24D4, C42.111D10, C10.1042+ 1+4, (C4×D4)⋊15D5, (D4×C20)⋊17C2, C4⋊1(C4○D20), C20⋊3(C4○D4), C20⋊D4⋊9C2, C5⋊1(Q8⋊6D4), C4.142(D4×D5), D10⋊D4⋊8C2, C20⋊7D4⋊19C2, C20⋊4D4⋊12C2, C4⋊C4.317D10, C20.348(C2×D4), D20⋊8C4⋊15C2, (C4×Dic10)⋊32C2, (C2×D4).216D10, (C2×C10).97C24, Dic5.42(C2×D4), C10.52(C22×D4), (C2×C20).785C23, (C4×C20).154C22, C22⋊C4.112D10, (C22×C4).210D10, C2.16(D4⋊8D10), C23.97(C22×D5), (D4×C10).258C22, (C2×D20).144C22, C4⋊Dic5.299C22, (C4×Dic5).83C22, (C22×D5).32C23, C22.122(C23×D5), D10⋊C4.54C22, (C22×C10).167C23, (C22×C20).109C22, (C2×Dic5).215C23, (C2×Dic10).324C22, C10.D4.111C22, C2.25(C2×D4×D5), (C2×C4○D20)⋊10C2, C10.44(C2×C4○D4), C2.48(C2×C4○D20), (C2×C4×D5).251C22, (C5×C4⋊C4).328C22, (C2×C4).580(C22×D5), (C2×C5⋊D4).14C22, (C5×C22⋊C4).124C22, SmallGroup(320,1225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊24D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 1270 in 312 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, Q8⋊6D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D4×C10, C4×Dic10, C20⋊4D4, D10⋊D4, D20⋊8C4, C20⋊7D4, C20⋊D4, D4×C20, C2×C4○D20, Dic10⋊24D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, Q8⋊6D4, C4○D20, D4×D5, C23×D5, C2×C4○D20, C2×D4×D5, D4⋊8D10, Dic10⋊24D4
►On 160 points
Generators in S160
(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 26 11 36)(2 25 12 35)(3 24 13 34)(4 23 14 33)(5 22 15 32)(6 21 16 31)(7 40 17 30)(8 39 18 29)(9 38 19 28)(10 37 20 27)(41 149 51 159)(42 148 52 158)(43 147 53 157)(44 146 54 156)(45 145 55 155)(46 144 56 154)(47 143 57 153)(48 142 58 152)(49 141 59 151)(50 160 60 150)(61 102 71 112)(62 101 72 111)(63 120 73 110)(64 119 74 109)(65 118 75 108)(66 117 76 107)(67 116 77 106)(68 115 78 105)(69 114 79 104)(70 113 80 103)(81 131 91 121)(82 130 92 140)(83 129 93 139)(84 128 94 138)(85 127 95 137)(86 126 96 136)(87 125 97 135)(88 124 98 134)(89 123 99 133)(90 122 100 132)
(1 158 84 78)(2 159 85 79)(3 160 86 80)(4 141 87 61)(5 142 88 62)(6 143 89 63)(7 144 90 64)(8 145 91 65)(9 146 92 66)(10 147 93 67)(11 148 94 68)(12 149 95 69)(13 150 96 70)(14 151 97 71)(15 152 98 72)(16 153 99 73)(17 154 100 74)(18 155 81 75)(19 156 82 76)(20 157 83 77)(21 57 123 120)(22 58 124 101)(23 59 125 102)(24 60 126 103)(25 41 127 104)(26 42 128 105)(27 43 129 106)(28 44 130 107)(29 45 131 108)(30 46 132 109)(31 47 133 110)(32 48 134 111)(33 49 135 112)(34 50 136 113)(35 51 137 114)(36 52 138 115)(37 53 139 116)(38 54 140 117)(39 55 121 118)(40 56 122 119)
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 121)(15 122)(16 123)(17 124)(18 125)(19 126)(20 127)(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 157)(42 158)(43 159)(44 160)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 108)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 120)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)
G:=sub<Sym(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), (1,26,11,36)(2,25,12,35)(3,24,13,34)(4,23,14,33)(5,22,15,32)(6,21,16,31)(7,40,17,30)(8,39,18,29)(9,38,19,28)(10,37,20,27)(41,149,51,159)(42,148,52,158)(43,147,53,157)(44,146,54,156)(45,145,55,155)(46,144,56,154)(47,143,57,153)(48,142,58,152)(49,141,59,151)(50,160,60,150)(61,102,71,112)(62,101,72,111)(63,120,73,110)(64,119,74,109)(65,118,75,108)(66,117,76,107)(67,116,77,106)(68,115,78,105)(69,114,79,104)(70,113,80,103)(81,131,91,121)(82,130,92,140)(83,129,93,139)(84,128,94,138)(85,127,95,137)(86,126,96,136)(87,125,97,135)(88,124,98,134)(89,123,99,133)(90,122,100,132), (1,158,84,78)(2,159,85,79)(3,160,86,80)(4,141,87,61)(5,142,88,62)(6,143,89,63)(7,144,90,64)(8,145,91,65)(9,146,92,66)(10,147,93,67)(11,148,94,68)(12,149,95,69)(13,150,96,70)(14,151,97,71)(15,152,98,72)(16,153,99,73)(17,154,100,74)(18,155,81,75)(19,156,82,76)(20,157,83,77)(21,57,123,120)(22,58,124,101)(23,59,125,102)(24,60,126,103)(25,41,127,104)(26,42,128,105)(27,43,129,106)(28,44,130,107)(29,45,131,108)(30,46,132,109)(31,47,133,110)(32,48,134,111)(33,49,135,112)(34,50,136,113)(35,51,137,114)(36,52,138,115)(37,53,139,116)(38,54,140,117)(39,55,121,118)(40,56,122,119), (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,157)(42,158)(43,159)(44,160)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,108)(62,109)(63,110)(64,111)(65,112)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119)(73,120)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)>;
G:=Group( (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), (1,26,11,36)(2,25,12,35)(3,24,13,34)(4,23,14,33)(5,22,15,32)(6,21,16,31)(7,40,17,30)(8,39,18,29)(9,38,19,28)(10,37,20,27)(41,149,51,159)(42,148,52,158)(43,147,53,157)(44,146,54,156)(45,145,55,155)(46,144,56,154)(47,143,57,153)(48,142,58,152)(49,141,59,151)(50,160,60,150)(61,102,71,112)(62,101,72,111)(63,120,73,110)(64,119,74,109)(65,118,75,108)(66,117,76,107)(67,116,77,106)(68,115,78,105)(69,114,79,104)(70,113,80,103)(81,131,91,121)(82,130,92,140)(83,129,93,139)(84,128,94,138)(85,127,95,137)(86,126,96,136)(87,125,97,135)(88,124,98,134)(89,123,99,133)(90,122,100,132), (1,158,84,78)(2,159,85,79)(3,160,86,80)(4,141,87,61)(5,142,88,62)(6,143,89,63)(7,144,90,64)(8,145,91,65)(9,146,92,66)(10,147,93,67)(11,148,94,68)(12,149,95,69)(13,150,96,70)(14,151,97,71)(15,152,98,72)(16,153,99,73)(17,154,100,74)(18,155,81,75)(19,156,82,76)(20,157,83,77)(21,57,123,120)(22,58,124,101)(23,59,125,102)(24,60,126,103)(25,41,127,104)(26,42,128,105)(27,43,129,106)(28,44,130,107)(29,45,131,108)(30,46,132,109)(31,47,133,110)(32,48,134,111)(33,49,135,112)(34,50,136,113)(35,51,137,114)(36,52,138,115)(37,53,139,116)(38,54,140,117)(39,55,121,118)(40,56,122,119), (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,157)(42,158)(43,159)(44,160)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,108)(62,109)(63,110)(64,111)(65,112)(66,113)(67,114)(68,115)(69,116)(70,117)(71,118)(72,119)(73,120)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107) );
G=PermutationGroup([[(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)], [(1,26,11,36),(2,25,12,35),(3,24,13,34),(4,23,14,33),(5,22,15,32),(6,21,16,31),(7,40,17,30),(8,39,18,29),(9,38,19,28),(10,37,20,27),(41,149,51,159),(42,148,52,158),(43,147,53,157),(44,146,54,156),(45,145,55,155),(46,144,56,154),(47,143,57,153),(48,142,58,152),(49,141,59,151),(50,160,60,150),(61,102,71,112),(62,101,72,111),(63,120,73,110),(64,119,74,109),(65,118,75,108),(66,117,76,107),(67,116,77,106),(68,115,78,105),(69,114,79,104),(70,113,80,103),(81,131,91,121),(82,130,92,140),(83,129,93,139),(84,128,94,138),(85,127,95,137),(86,126,96,136),(87,125,97,135),(88,124,98,134),(89,123,99,133),(90,122,100,132)], [(1,158,84,78),(2,159,85,79),(3,160,86,80),(4,141,87,61),(5,142,88,62),(6,143,89,63),(7,144,90,64),(8,145,91,65),(9,146,92,66),(10,147,93,67),(11,148,94,68),(12,149,95,69),(13,150,96,70),(14,151,97,71),(15,152,98,72),(16,153,99,73),(17,154,100,74),(18,155,81,75),(19,156,82,76),(20,157,83,77),(21,57,123,120),(22,58,124,101),(23,59,125,102),(24,60,126,103),(25,41,127,104),(26,42,128,105),(27,43,129,106),(28,44,130,107),(29,45,131,108),(30,46,132,109),(31,47,133,110),(32,48,134,111),(33,49,135,112),(34,50,136,113),(35,51,137,114),(36,52,138,115),(37,53,139,116),(38,54,140,117),(39,55,121,118),(40,56,122,119)], [(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,136),(10,137),(11,138),(12,139),(13,140),(14,121),(15,122),(16,123),(17,124),(18,125),(19,126),(20,127),(21,99),(22,100),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,157),(42,158),(43,159),(44,160),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,108),(62,109),(63,110),(64,111),(65,112),(66,113),(67,114),(68,115),(69,116),(70,117),(71,118),(72,119),(73,120),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4×D5 | D4⋊8D10 |
| kernel | Dic10⋊24D4 | C4×Dic10 | C20⋊4D4 | D10⋊D4 | D20⋊8C4 | C20⋊7D4 | C20⋊D4 | D4×C20 | C2×C4○D20 | Dic10 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of Dic10⋊24D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 20 | 0 | 0 |
| 0 | 0 | 18 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 16 | 0 | 0 |
| 0 | 0 | 3 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 36 |
| 0 | 0 | 0 | 0 | 10 | 34 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 25 | 0 | 0 |
| 0 | 0 | 2 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 11 | 40 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,7,18,0,0,0,0,20,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,22,3,0,0,0,0,16,19,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,0,0,0,36,34],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,19,2,0,0,0,0,25,22,0,0,0,0,0,0,1,11,0,0,0,0,0,40] >;
Dic10⋊24D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{24}D_4 % in TeX
G:=Group("Dic10:24D4"); // GroupNames label
G:=SmallGroup(320,1225);
// by ID
G=gap.SmallGroup(320,1225);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations