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 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×10], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×7], Dic5 [×5], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, C40 [×2], Dic10 [×4], Dic10 [×6], D20 [×2], C2×Dic5 [×7], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, Q8⋊D4, C40⋊C2 [×4], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×Dic10 [×2], C2×Dic10 [×5], C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C20.44D4 [×2], C5×C22⋊C8, C2×C40⋊C2 [×2], C20⋊7D4, C22×Dic10, Dic10⋊14D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8.C22, D20 [×2], C22×D5, Q8⋊D4, C40⋊C2 [×2], C2×D20, D4×D5 [×2], 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)
(1 155 11 145)(2 154 12 144)(3 153 13 143)(4 152 14 142)(5 151 15 141)(6 150 16 160)(7 149 17 159)(8 148 18 158)(9 147 19 157)(10 146 20 156)(21 46 31 56)(22 45 32 55)(23 44 33 54)(24 43 34 53)(25 42 35 52)(26 41 36 51)(27 60 37 50)(28 59 38 49)(29 58 39 48)(30 57 40 47)(61 105 71 115)(62 104 72 114)(63 103 73 113)(64 102 74 112)(65 101 75 111)(66 120 76 110)(67 119 77 109)(68 118 78 108)(69 117 79 107)(70 116 80 106)(81 132 91 122)(82 131 92 121)(83 130 93 140)(84 129 94 139)(85 128 95 138)(86 127 96 137)(87 126 97 136)(88 125 98 135)(89 124 99 134)(90 123 100 133)
(1 36 107 81)(2 35 108 100)(3 34 109 99)(4 33 110 98)(5 32 111 97)(6 31 112 96)(7 30 113 95)(8 29 114 94)(9 28 115 93)(10 27 116 92)(11 26 117 91)(12 25 118 90)(13 24 119 89)(14 23 120 88)(15 22 101 87)(16 21 102 86)(17 40 103 85)(18 39 104 84)(19 38 105 83)(20 37 106 82)(41 74 122 160)(42 73 123 159)(43 72 124 158)(44 71 125 157)(45 70 126 156)(46 69 127 155)(47 68 128 154)(48 67 129 153)(49 66 130 152)(50 65 131 151)(51 64 132 150)(52 63 133 149)(53 62 134 148)(54 61 135 147)(55 80 136 146)(56 79 137 145)(57 78 138 144)(58 77 139 143)(59 76 140 142)(60 75 121 141)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 141)(58 142)(59 143)(60 144)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 121)(79 122)(80 123)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)
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), (1,155,11,145)(2,154,12,144)(3,153,13,143)(4,152,14,142)(5,151,15,141)(6,150,16,160)(7,149,17,159)(8,148,18,158)(9,147,19,157)(10,146,20,156)(21,46,31,56)(22,45,32,55)(23,44,33,54)(24,43,34,53)(25,42,35,52)(26,41,36,51)(27,60,37,50)(28,59,38,49)(29,58,39,48)(30,57,40,47)(61,105,71,115)(62,104,72,114)(63,103,73,113)(64,102,74,112)(65,101,75,111)(66,120,76,110)(67,119,77,109)(68,118,78,108)(69,117,79,107)(70,116,80,106)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133), (1,36,107,81)(2,35,108,100)(3,34,109,99)(4,33,110,98)(5,32,111,97)(6,31,112,96)(7,30,113,95)(8,29,114,94)(9,28,115,93)(10,27,116,92)(11,26,117,91)(12,25,118,90)(13,24,119,89)(14,23,120,88)(15,22,101,87)(16,21,102,86)(17,40,103,85)(18,39,104,84)(19,38,105,83)(20,37,106,82)(41,74,122,160)(42,73,123,159)(43,72,124,158)(44,71,125,157)(45,70,126,156)(46,69,127,155)(47,68,128,154)(48,67,129,153)(49,66,130,152)(50,65,131,151)(51,64,132,150)(52,63,133,149)(53,62,134,148)(54,61,135,147)(55,80,136,146)(56,79,137,145)(57,78,138,144)(58,77,139,143)(59,76,140,142)(60,75,121,141), (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,121)(79,122)(80,123)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)>;
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), (1,155,11,145)(2,154,12,144)(3,153,13,143)(4,152,14,142)(5,151,15,141)(6,150,16,160)(7,149,17,159)(8,148,18,158)(9,147,19,157)(10,146,20,156)(21,46,31,56)(22,45,32,55)(23,44,33,54)(24,43,34,53)(25,42,35,52)(26,41,36,51)(27,60,37,50)(28,59,38,49)(29,58,39,48)(30,57,40,47)(61,105,71,115)(62,104,72,114)(63,103,73,113)(64,102,74,112)(65,101,75,111)(66,120,76,110)(67,119,77,109)(68,118,78,108)(69,117,79,107)(70,116,80,106)(81,132,91,122)(82,131,92,121)(83,130,93,140)(84,129,94,139)(85,128,95,138)(86,127,96,137)(87,126,97,136)(88,125,98,135)(89,124,99,134)(90,123,100,133), (1,36,107,81)(2,35,108,100)(3,34,109,99)(4,33,110,98)(5,32,111,97)(6,31,112,96)(7,30,113,95)(8,29,114,94)(9,28,115,93)(10,27,116,92)(11,26,117,91)(12,25,118,90)(13,24,119,89)(14,23,120,88)(15,22,101,87)(16,21,102,86)(17,40,103,85)(18,39,104,84)(19,38,105,83)(20,37,106,82)(41,74,122,160)(42,73,123,159)(43,72,124,158)(44,71,125,157)(45,70,126,156)(46,69,127,155)(47,68,128,154)(48,67,129,153)(49,66,130,152)(50,65,131,151)(51,64,132,150)(52,63,133,149)(53,62,134,148)(54,61,135,147)(55,80,136,146)(56,79,137,145)(57,78,138,144)(58,77,139,143)(59,76,140,142)(60,75,121,141), (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,121)(79,122)(80,123)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106) );
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)], [(1,155,11,145),(2,154,12,144),(3,153,13,143),(4,152,14,142),(5,151,15,141),(6,150,16,160),(7,149,17,159),(8,148,18,158),(9,147,19,157),(10,146,20,156),(21,46,31,56),(22,45,32,55),(23,44,33,54),(24,43,34,53),(25,42,35,52),(26,41,36,51),(27,60,37,50),(28,59,38,49),(29,58,39,48),(30,57,40,47),(61,105,71,115),(62,104,72,114),(63,103,73,113),(64,102,74,112),(65,101,75,111),(66,120,76,110),(67,119,77,109),(68,118,78,108),(69,117,79,107),(70,116,80,106),(81,132,91,122),(82,131,92,121),(83,130,93,140),(84,129,94,139),(85,128,95,138),(86,127,96,137),(87,126,97,136),(88,125,98,135),(89,124,99,134),(90,123,100,133)], [(1,36,107,81),(2,35,108,100),(3,34,109,99),(4,33,110,98),(5,32,111,97),(6,31,112,96),(7,30,113,95),(8,29,114,94),(9,28,115,93),(10,27,116,92),(11,26,117,91),(12,25,118,90),(13,24,119,89),(14,23,120,88),(15,22,101,87),(16,21,102,86),(17,40,103,85),(18,39,104,84),(19,38,105,83),(20,37,106,82),(41,74,122,160),(42,73,123,159),(43,72,124,158),(44,71,125,157),(45,70,126,156),(46,69,127,155),(47,68,128,154),(48,67,129,153),(49,66,130,152),(50,65,131,151),(51,64,132,150),(52,63,133,149),(53,62,134,148),(54,61,135,147),(55,80,136,146),(56,79,137,145),(57,78,138,144),(58,77,139,143),(59,76,140,142),(60,75,121,141)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,141),(58,142),(59,143),(60,144),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,121),(79,122),(80,123),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106)])
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