metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊24D4, C42.111D10, C10.1042+ (1+4), (C4×D4)⋊15D5, (D4×C20)⋊17C2, C20⋊3(C4○D4), C4⋊1(C4○D20), C20⋊D4⋊9C2, C5⋊1(Q8⋊6D4), C4.142(D4×D5), D10⋊D4⋊8C2, C4⋊D20⋊12C2, C20⋊7D4⋊19C2, 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), (C2×D20).144C22, (D4×C10).258C22, 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, C2.48(C2×C4○D20), C10.44(C2×C4○D4), (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
Subgroups: 1270 in 312 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×18], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×24], Q8 [×4], C23 [×2], C23 [×4], D5 [×4], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×14], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×2], C20 [×4], C20 [×3], D10 [×12], C2×C10, C2×C10 [×6], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×8], D20 [×10], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5 [×4], C22×C10 [×2], Q8⋊6D4, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×4], C2×D20 [×6], C4○D20 [×8], C2×C5⋊D4 [×8], C22×C20 [×2], D4×C10, C4×Dic10, C4⋊D20, D10⋊D4 [×4], D20⋊8C4 [×2], C20⋊7D4 [×2], C20⋊D4 [×2], D4×C20, C2×C4○D20 [×2], Dic10⋊24D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D5 [×7], Q8⋊6D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4⋊8D10, Dic10⋊24D4
Generators and relations
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 >
►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 67 11 77)(2 66 12 76)(3 65 13 75)(4 64 14 74)(5 63 15 73)(6 62 16 72)(7 61 17 71)(8 80 18 70)(9 79 19 69)(10 78 20 68)(21 133 31 123)(22 132 32 122)(23 131 33 121)(24 130 34 140)(25 129 35 139)(26 128 36 138)(27 127 37 137)(28 126 38 136)(29 125 39 135)(30 124 40 134)(41 87 51 97)(42 86 52 96)(43 85 53 95)(44 84 54 94)(45 83 55 93)(46 82 56 92)(47 81 57 91)(48 100 58 90)(49 99 59 89)(50 98 60 88)(101 152 111 142)(102 151 112 141)(103 150 113 160)(104 149 114 159)(105 148 115 158)(106 147 116 157)(107 146 117 156)(108 145 118 155)(109 144 119 154)(110 143 120 153)
(1 129 60 107)(2 130 41 108)(3 131 42 109)(4 132 43 110)(5 133 44 111)(6 134 45 112)(7 135 46 113)(8 136 47 114)(9 137 48 115)(10 138 49 116)(11 139 50 117)(12 140 51 118)(13 121 52 119)(14 122 53 120)(15 123 54 101)(16 124 55 102)(17 125 56 103)(18 126 57 104)(19 127 58 105)(20 128 59 106)(21 94 152 73)(22 95 153 74)(23 96 154 75)(24 97 155 76)(25 98 156 77)(26 99 157 78)(27 100 158 79)(28 81 159 80)(29 82 160 61)(30 83 141 62)(31 84 142 63)(32 85 143 64)(33 86 144 65)(34 87 145 66)(35 88 146 67)(36 89 147 68)(37 90 148 69)(38 91 149 70)(39 92 150 71)(40 93 151 72)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(101 160)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)
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), (1,67,11,77)(2,66,12,76)(3,65,13,75)(4,64,14,74)(5,63,15,73)(6,62,16,72)(7,61,17,71)(8,80,18,70)(9,79,19,69)(10,78,20,68)(21,133,31,123)(22,132,32,122)(23,131,33,121)(24,130,34,140)(25,129,35,139)(26,128,36,138)(27,127,37,137)(28,126,38,136)(29,125,39,135)(30,124,40,134)(41,87,51,97)(42,86,52,96)(43,85,53,95)(44,84,54,94)(45,83,55,93)(46,82,56,92)(47,81,57,91)(48,100,58,90)(49,99,59,89)(50,98,60,88)(101,152,111,142)(102,151,112,141)(103,150,113,160)(104,149,114,159)(105,148,115,158)(106,147,116,157)(107,146,117,156)(108,145,118,155)(109,144,119,154)(110,143,120,153), (1,129,60,107)(2,130,41,108)(3,131,42,109)(4,132,43,110)(5,133,44,111)(6,134,45,112)(7,135,46,113)(8,136,47,114)(9,137,48,115)(10,138,49,116)(11,139,50,117)(12,140,51,118)(13,121,52,119)(14,122,53,120)(15,123,54,101)(16,124,55,102)(17,125,56,103)(18,126,57,104)(19,127,58,105)(20,128,59,106)(21,94,152,73)(22,95,153,74)(23,96,154,75)(24,97,155,76)(25,98,156,77)(26,99,157,78)(27,100,158,79)(28,81,159,80)(29,82,160,61)(30,83,141,62)(31,84,142,63)(32,85,143,64)(33,86,144,65)(34,87,145,66)(35,88,146,67)(36,89,147,68)(37,90,148,69)(38,91,149,70)(39,92,150,71)(40,93,151,72), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(101,160)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159)>;
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), (1,67,11,77)(2,66,12,76)(3,65,13,75)(4,64,14,74)(5,63,15,73)(6,62,16,72)(7,61,17,71)(8,80,18,70)(9,79,19,69)(10,78,20,68)(21,133,31,123)(22,132,32,122)(23,131,33,121)(24,130,34,140)(25,129,35,139)(26,128,36,138)(27,127,37,137)(28,126,38,136)(29,125,39,135)(30,124,40,134)(41,87,51,97)(42,86,52,96)(43,85,53,95)(44,84,54,94)(45,83,55,93)(46,82,56,92)(47,81,57,91)(48,100,58,90)(49,99,59,89)(50,98,60,88)(101,152,111,142)(102,151,112,141)(103,150,113,160)(104,149,114,159)(105,148,115,158)(106,147,116,157)(107,146,117,156)(108,145,118,155)(109,144,119,154)(110,143,120,153), (1,129,60,107)(2,130,41,108)(3,131,42,109)(4,132,43,110)(5,133,44,111)(6,134,45,112)(7,135,46,113)(8,136,47,114)(9,137,48,115)(10,138,49,116)(11,139,50,117)(12,140,51,118)(13,121,52,119)(14,122,53,120)(15,123,54,101)(16,124,55,102)(17,125,56,103)(18,126,57,104)(19,127,58,105)(20,128,59,106)(21,94,152,73)(22,95,153,74)(23,96,154,75)(24,97,155,76)(25,98,156,77)(26,99,157,78)(27,100,158,79)(28,81,159,80)(29,82,160,61)(30,83,141,62)(31,84,142,63)(32,85,143,64)(33,86,144,65)(34,87,145,66)(35,88,146,67)(36,89,147,68)(37,90,148,69)(38,91,149,70)(39,92,150,71)(40,93,151,72), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67)(101,160)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159) );
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)], [(1,67,11,77),(2,66,12,76),(3,65,13,75),(4,64,14,74),(5,63,15,73),(6,62,16,72),(7,61,17,71),(8,80,18,70),(9,79,19,69),(10,78,20,68),(21,133,31,123),(22,132,32,122),(23,131,33,121),(24,130,34,140),(25,129,35,139),(26,128,36,138),(27,127,37,137),(28,126,38,136),(29,125,39,135),(30,124,40,134),(41,87,51,97),(42,86,52,96),(43,85,53,95),(44,84,54,94),(45,83,55,93),(46,82,56,92),(47,81,57,91),(48,100,58,90),(49,99,59,89),(50,98,60,88),(101,152,111,142),(102,151,112,141),(103,150,113,160),(104,149,114,159),(105,148,115,158),(106,147,116,157),(107,146,117,156),(108,145,118,155),(109,144,119,154),(110,143,120,153)], [(1,129,60,107),(2,130,41,108),(3,131,42,109),(4,132,43,110),(5,133,44,111),(6,134,45,112),(7,135,46,113),(8,136,47,114),(9,137,48,115),(10,138,49,116),(11,139,50,117),(12,140,51,118),(13,121,52,119),(14,122,53,120),(15,123,54,101),(16,124,55,102),(17,125,56,103),(18,126,57,104),(19,127,58,105),(20,128,59,106),(21,94,152,73),(22,95,153,74),(23,96,154,75),(24,97,155,76),(25,98,156,77),(26,99,157,78),(27,100,158,79),(28,81,159,80),(29,82,160,61),(30,83,141,62),(31,84,142,63),(32,85,143,64),(33,86,144,65),(34,87,145,66),(35,88,146,67),(36,89,147,68),(37,90,148,69),(38,91,149,70),(39,92,150,71),(40,93,151,72)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67),(101,160),(102,141),(103,142),(104,143),(105,144),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,157),(119,158),(120,159)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D20 | 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 |
In GAP, Magma, Sage, TeX
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
×
⋊
ℤ
𝔽
○
≀