metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊14D4, C23.40D20, C22⋊C8⋊10D5, C5⋊1(Q8⋊D4), (C2×C10)⋊2SD16, C4.123(D4×D5), (C2×C4).35D20, (C2×C20).46D4, C20.335(C2×D4), C20⋊7D4.4C2, (C2×C8).111D10, C10.9(C2×SD16), C10.11C22≀C2, C22⋊2(C40⋊C2), (C22×C4).88D10, (C22×C10).58D4, C20.44D4⋊11C2, (C2×C40).122C22, (C2×C20).748C23, (C22×Dic10)⋊2C2, (C2×D20).15C22, C22.111(C2×D20), C4⋊Dic5.15C22, C2.14(C22⋊D20), C2.14(C8.D10), C10.11(C8.C22), (C22×C20).54C22, (C2×Dic10).218C22, (C2×C40⋊C2)⋊12C2, (C5×C22⋊C8)⋊12C2, C2.12(C2×C40⋊C2), (C2×C10).131(C2×D4), (C2×C4).693(C22×D5), SmallGroup(320,365)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊14D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >
Subgroups: 734 in 158 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C40, Dic10, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Q8⋊D4, C40⋊C2, C4⋊Dic5, D10⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C20.44D4, C5×C22⋊C8, C2×C40⋊C2, C20⋊7D4, C22×Dic10, Dic10⋊14D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8.C22, D20, C22×D5, Q8⋊D4, C40⋊C2, C2×D20, D4×D5, C22⋊D20, C2×C40⋊C2, C8.D10, Dic10⋊14D4
►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)
(1 142 11 152)(2 141 12 151)(3 160 13 150)(4 159 14 149)(5 158 15 148)(6 157 16 147)(7 156 17 146)(8 155 18 145)(9 154 19 144)(10 153 20 143)(21 88 31 98)(22 87 32 97)(23 86 33 96)(24 85 34 95)(25 84 35 94)(26 83 36 93)(27 82 37 92)(28 81 38 91)(29 100 39 90)(30 99 40 89)(41 105 51 115)(42 104 52 114)(43 103 53 113)(44 102 54 112)(45 101 55 111)(46 120 56 110)(47 119 57 109)(48 118 58 108)(49 117 59 107)(50 116 60 106)(61 125 71 135)(62 124 72 134)(63 123 73 133)(64 122 74 132)(65 121 75 131)(66 140 76 130)(67 139 77 129)(68 138 78 128)(69 137 79 127)(70 136 80 126)
(1 79 85 118)(2 78 86 117)(3 77 87 116)(4 76 88 115)(5 75 89 114)(6 74 90 113)(7 73 91 112)(8 72 92 111)(9 71 93 110)(10 70 94 109)(11 69 95 108)(12 68 96 107)(13 67 97 106)(14 66 98 105)(15 65 99 104)(16 64 100 103)(17 63 81 102)(18 62 82 101)(19 61 83 120)(20 80 84 119)(21 46 149 135)(22 45 150 134)(23 44 151 133)(24 43 152 132)(25 42 153 131)(26 41 154 130)(27 60 155 129)(28 59 156 128)(29 58 157 127)(30 57 158 126)(31 56 159 125)(32 55 160 124)(33 54 141 123)(34 53 142 122)(35 52 143 121)(36 51 144 140)(37 50 145 139)(38 49 146 138)(39 48 147 137)(40 47 148 136)
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
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), (1,142,11,152)(2,141,12,151)(3,160,13,150)(4,159,14,149)(5,158,15,148)(6,157,16,147)(7,156,17,146)(8,155,18,145)(9,154,19,144)(10,153,20,143)(21,88,31,98)(22,87,32,97)(23,86,33,96)(24,85,34,95)(25,84,35,94)(26,83,36,93)(27,82,37,92)(28,81,38,91)(29,100,39,90)(30,99,40,89)(41,105,51,115)(42,104,52,114)(43,103,53,113)(44,102,54,112)(45,101,55,111)(46,120,56,110)(47,119,57,109)(48,118,58,108)(49,117,59,107)(50,116,60,106)(61,125,71,135)(62,124,72,134)(63,123,73,133)(64,122,74,132)(65,121,75,131)(66,140,76,130)(67,139,77,129)(68,138,78,128)(69,137,79,127)(70,136,80,126), (1,79,85,118)(2,78,86,117)(3,77,87,116)(4,76,88,115)(5,75,89,114)(6,74,90,113)(7,73,91,112)(8,72,92,111)(9,71,93,110)(10,70,94,109)(11,69,95,108)(12,68,96,107)(13,67,97,106)(14,66,98,105)(15,65,99,104)(16,64,100,103)(17,63,81,102)(18,62,82,101)(19,61,83,120)(20,80,84,119)(21,46,149,135)(22,45,150,134)(23,44,151,133)(24,43,152,132)(25,42,153,131)(26,41,154,130)(27,60,155,129)(28,59,156,128)(29,58,157,127)(30,57,158,126)(31,56,159,125)(32,55,160,124)(33,54,141,123)(34,53,142,122)(35,52,143,121)(36,51,144,140)(37,50,145,139)(38,49,146,138)(39,48,147,137)(40,47,148,136), (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,41)(38,42)(39,43)(40,44)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)>;
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), (1,142,11,152)(2,141,12,151)(3,160,13,150)(4,159,14,149)(5,158,15,148)(6,157,16,147)(7,156,17,146)(8,155,18,145)(9,154,19,144)(10,153,20,143)(21,88,31,98)(22,87,32,97)(23,86,33,96)(24,85,34,95)(25,84,35,94)(26,83,36,93)(27,82,37,92)(28,81,38,91)(29,100,39,90)(30,99,40,89)(41,105,51,115)(42,104,52,114)(43,103,53,113)(44,102,54,112)(45,101,55,111)(46,120,56,110)(47,119,57,109)(48,118,58,108)(49,117,59,107)(50,116,60,106)(61,125,71,135)(62,124,72,134)(63,123,73,133)(64,122,74,132)(65,121,75,131)(66,140,76,130)(67,139,77,129)(68,138,78,128)(69,137,79,127)(70,136,80,126), (1,79,85,118)(2,78,86,117)(3,77,87,116)(4,76,88,115)(5,75,89,114)(6,74,90,113)(7,73,91,112)(8,72,92,111)(9,71,93,110)(10,70,94,109)(11,69,95,108)(12,68,96,107)(13,67,97,106)(14,66,98,105)(15,65,99,104)(16,64,100,103)(17,63,81,102)(18,62,82,101)(19,61,83,120)(20,80,84,119)(21,46,149,135)(22,45,150,134)(23,44,151,133)(24,43,152,132)(25,42,153,131)(26,41,154,130)(27,60,155,129)(28,59,156,128)(29,58,157,127)(30,57,158,126)(31,56,159,125)(32,55,160,124)(33,54,141,123)(34,53,142,122)(35,52,143,121)(36,51,144,140)(37,50,145,139)(38,49,146,138)(39,48,147,137)(40,47,148,136), (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,41)(38,42)(39,43)(40,44)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155) );
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)], [(1,142,11,152),(2,141,12,151),(3,160,13,150),(4,159,14,149),(5,158,15,148),(6,157,16,147),(7,156,17,146),(8,155,18,145),(9,154,19,144),(10,153,20,143),(21,88,31,98),(22,87,32,97),(23,86,33,96),(24,85,34,95),(25,84,35,94),(26,83,36,93),(27,82,37,92),(28,81,38,91),(29,100,39,90),(30,99,40,89),(41,105,51,115),(42,104,52,114),(43,103,53,113),(44,102,54,112),(45,101,55,111),(46,120,56,110),(47,119,57,109),(48,118,58,108),(49,117,59,107),(50,116,60,106),(61,125,71,135),(62,124,72,134),(63,123,73,133),(64,122,74,132),(65,121,75,131),(66,140,76,130),(67,139,77,129),(68,138,78,128),(69,137,79,127),(70,136,80,126)], [(1,79,85,118),(2,78,86,117),(3,77,87,116),(4,76,88,115),(5,75,89,114),(6,74,90,113),(7,73,91,112),(8,72,92,111),(9,71,93,110),(10,70,94,109),(11,69,95,108),(12,68,96,107),(13,67,97,106),(14,66,98,105),(15,65,99,104),(16,64,100,103),(17,63,81,102),(18,62,82,101),(19,61,83,120),(20,80,84,119),(21,46,149,135),(22,45,150,134),(23,44,151,133),(24,43,152,132),(25,42,153,131),(26,41,154,130),(27,60,155,129),(28,59,156,128),(29,58,157,127),(30,57,158,126),(31,56,159,125),(32,55,160,124),(33,54,141,123),(34,53,142,122),(35,52,143,121),(36,51,144,140),(37,50,145,139),(38,49,146,138),(39,48,147,137),(40,47,148,136)], [(1,79),(2,80),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,41),(38,42),(39,43),(40,44),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,101),(89,102),(90,103),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,111),(99,112),(100,113),(121,156),(122,157),(123,158),(124,159),(125,160),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 | C8.C22 | D4×D5 | C8.D10 |
| kernel | Dic10⋊14D4 | C20.44D4 | C5×C22⋊C8 | C2×C40⋊C2 | C20⋊7D4 | C22×Dic10 | Dic10 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of Dic10⋊14D4 ►in GL4(𝔽41) generated by
| 13 | 2 | 0 | 0 |
| 39 | 25 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 15 | 31 | 0 | 0 |
| 39 | 26 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 39 | 28 | 0 | 0 |
| 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [13,39,0,0,2,25,0,0,0,0,40,0,0,0,0,40],[15,39,0,0,31,26,0,0,0,0,0,40,0,0,40,0],[39,16,0,0,28,2,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;
Dic10⋊14D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_{14}D_4 % in TeX
G:=Group("Dic10:14D4"); // GroupNames label
G:=SmallGroup(320,365);
// by ID
G=gap.SmallGroup(320,365);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations