direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×Dic10, C42.101D10, C10.552- (1+4), C5⋊1(D4×Q8), (C5×D4)⋊5Q8, C20⋊1(C2×Q8), C20⋊Q8⋊14C2, (C4×D4).10D5, C4⋊1(C2×Dic10), C4.138(D4×D5), C4⋊C4.276D10, C20⋊2Q8⋊21C2, (D4×C20).11C2, C20.344(C2×D4), (C4×Dic10)⋊24C2, (C2×D4).241D10, C20.48D4⋊6C2, (C2×C10).81C24, (D4×Dic5).11C2, Dic5.40(C2×D4), C22⋊1(C2×Dic10), C10.45(C22×D4), C10.12(C22×Q8), (C2×C20).153C23, (C4×C20).144C22, C22⋊C4.104D10, (C22×Dic10)⋊8C2, (C22×C4).201D10, Dic5.14D4⋊6C2, C23.D5.7C22, (D4×C10).248C22, C4⋊Dic5.197C22, (C22×C20).76C22, (C4×Dic5).79C22, C2.14(C22×Dic10), C10.D4.5C22, C22.109(C23×D5), C23.162(C22×D5), (C22×C10).151C23, (C2×Dic5).209C23, C2.13(D4.10D10), (C2×Dic10).242C22, (C22×Dic5).90C22, C2.18(C2×D4×D5), (C2×C10)⋊1(C2×Q8), (C5×C4⋊C4).317C22, (C2×C4).152(C22×D5), (C5×C22⋊C4).103C22, SmallGroup(320,1209)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 886 in 280 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×13], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×20], D4 [×4], Q8 [×16], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×11], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×15], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], Dic10 [×4], Dic10 [×12], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], D4×Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×6], C2×Dic10 [×8], C22×Dic5 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C20⋊2Q8, Dic5.14D4 [×4], C20⋊Q8 [×2], C20.48D4 [×2], D4×Dic5 [×2], D4×C20, C22×Dic10 [×2], D4×Dic10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- (1+4), Dic10 [×4], C22×D5 [×7], D4×Q8, C2×Dic10 [×6], D4×D5 [×2], C23×D5, C22×Dic10, C2×D4×D5, D4.10D10, D4×Dic10
Generators and relations
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 52 136 21)(2 53 137 22)(3 54 138 23)(4 55 139 24)(5 56 140 25)(6 57 121 26)(7 58 122 27)(8 59 123 28)(9 60 124 29)(10 41 125 30)(11 42 126 31)(12 43 127 32)(13 44 128 33)(14 45 129 34)(15 46 130 35)(16 47 131 36)(17 48 132 37)(18 49 133 38)(19 50 134 39)(20 51 135 40)(61 90 158 120)(62 91 159 101)(63 92 160 102)(64 93 141 103)(65 94 142 104)(66 95 143 105)(67 96 144 106)(68 97 145 107)(69 98 146 108)(70 99 147 109)(71 100 148 110)(72 81 149 111)(73 82 150 112)(74 83 151 113)(75 84 152 114)(76 85 153 115)(77 86 154 116)(78 87 155 117)(79 88 156 118)(80 89 157 119)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)
(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 81 11 91)(2 100 12 90)(3 99 13 89)(4 98 14 88)(5 97 15 87)(6 96 16 86)(7 95 17 85)(8 94 18 84)(9 93 19 83)(10 92 20 82)(21 72 31 62)(22 71 32 61)(23 70 33 80)(24 69 34 79)(25 68 35 78)(26 67 36 77)(27 66 37 76)(28 65 38 75)(29 64 39 74)(30 63 40 73)(41 160 51 150)(42 159 52 149)(43 158 53 148)(44 157 54 147)(45 156 55 146)(46 155 56 145)(47 154 57 144)(48 153 58 143)(49 152 59 142)(50 151 60 141)(101 136 111 126)(102 135 112 125)(103 134 113 124)(104 133 114 123)(105 132 115 122)(106 131 116 121)(107 130 117 140)(108 129 118 139)(109 128 119 138)(110 127 120 137)
G:=sub<Sym(160)| (1,52,136,21)(2,53,137,22)(3,54,138,23)(4,55,139,24)(5,56,140,25)(6,57,121,26)(7,58,122,27)(8,59,123,28)(9,60,124,29)(10,41,125,30)(11,42,126,31)(12,43,127,32)(13,44,128,33)(14,45,129,34)(15,46,130,35)(16,47,131,36)(17,48,132,37)(18,49,133,38)(19,50,134,39)(20,51,135,40)(61,90,158,120)(62,91,159,101)(63,92,160,102)(64,93,141,103)(65,94,142,104)(66,95,143,105)(67,96,144,106)(68,97,145,107)(69,98,146,108)(70,99,147,109)(71,100,148,110)(72,81,149,111)(73,82,150,112)(74,83,151,113)(75,84,152,114)(76,85,153,115)(77,86,154,116)(78,87,155,117)(79,88,156,118)(80,89,157,119), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148), (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), (1,81,11,91)(2,100,12,90)(3,99,13,89)(4,98,14,88)(5,97,15,87)(6,96,16,86)(7,95,17,85)(8,94,18,84)(9,93,19,83)(10,92,20,82)(21,72,31,62)(22,71,32,61)(23,70,33,80)(24,69,34,79)(25,68,35,78)(26,67,36,77)(27,66,37,76)(28,65,38,75)(29,64,39,74)(30,63,40,73)(41,160,51,150)(42,159,52,149)(43,158,53,148)(44,157,54,147)(45,156,55,146)(46,155,56,145)(47,154,57,144)(48,153,58,143)(49,152,59,142)(50,151,60,141)(101,136,111,126)(102,135,112,125)(103,134,113,124)(104,133,114,123)(105,132,115,122)(106,131,116,121)(107,130,117,140)(108,129,118,139)(109,128,119,138)(110,127,120,137)>;
G:=Group( (1,52,136,21)(2,53,137,22)(3,54,138,23)(4,55,139,24)(5,56,140,25)(6,57,121,26)(7,58,122,27)(8,59,123,28)(9,60,124,29)(10,41,125,30)(11,42,126,31)(12,43,127,32)(13,44,128,33)(14,45,129,34)(15,46,130,35)(16,47,131,36)(17,48,132,37)(18,49,133,38)(19,50,134,39)(20,51,135,40)(61,90,158,120)(62,91,159,101)(63,92,160,102)(64,93,141,103)(65,94,142,104)(66,95,143,105)(67,96,144,106)(68,97,145,107)(69,98,146,108)(70,99,147,109)(71,100,148,110)(72,81,149,111)(73,82,150,112)(74,83,151,113)(75,84,152,114)(76,85,153,115)(77,86,154,116)(78,87,155,117)(79,88,156,118)(80,89,157,119), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148), (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), (1,81,11,91)(2,100,12,90)(3,99,13,89)(4,98,14,88)(5,97,15,87)(6,96,16,86)(7,95,17,85)(8,94,18,84)(9,93,19,83)(10,92,20,82)(21,72,31,62)(22,71,32,61)(23,70,33,80)(24,69,34,79)(25,68,35,78)(26,67,36,77)(27,66,37,76)(28,65,38,75)(29,64,39,74)(30,63,40,73)(41,160,51,150)(42,159,52,149)(43,158,53,148)(44,157,54,147)(45,156,55,146)(46,155,56,145)(47,154,57,144)(48,153,58,143)(49,152,59,142)(50,151,60,141)(101,136,111,126)(102,135,112,125)(103,134,113,124)(104,133,114,123)(105,132,115,122)(106,131,116,121)(107,130,117,140)(108,129,118,139)(109,128,119,138)(110,127,120,137) );
G=PermutationGroup([(1,52,136,21),(2,53,137,22),(3,54,138,23),(4,55,139,24),(5,56,140,25),(6,57,121,26),(7,58,122,27),(8,59,123,28),(9,60,124,29),(10,41,125,30),(11,42,126,31),(12,43,127,32),(13,44,128,33),(14,45,129,34),(15,46,130,35),(16,47,131,36),(17,48,132,37),(18,49,133,38),(19,50,134,39),(20,51,135,40),(61,90,158,120),(62,91,159,101),(63,92,160,102),(64,93,141,103),(65,94,142,104),(66,95,143,105),(67,96,144,106),(68,97,145,107),(69,98,146,108),(70,99,147,109),(71,100,148,110),(72,81,149,111),(73,82,150,112),(74,83,151,113),(75,84,152,114),(76,85,153,115),(77,86,154,116),(78,87,155,117),(79,88,156,118),(80,89,157,119)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,100),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99),(101,149),(102,150),(103,151),(104,152),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,159),(112,160),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148)], [(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)], [(1,81,11,91),(2,100,12,90),(3,99,13,89),(4,98,14,88),(5,97,15,87),(6,96,16,86),(7,95,17,85),(8,94,18,84),(9,93,19,83),(10,92,20,82),(21,72,31,62),(22,71,32,61),(23,70,33,80),(24,69,34,79),(25,68,35,78),(26,67,36,77),(27,66,37,76),(28,65,38,75),(29,64,39,74),(30,63,40,73),(41,160,51,150),(42,159,52,149),(43,158,53,148),(44,157,54,147),(45,156,55,146),(46,155,56,145),(47,154,57,144),(48,153,58,143),(49,152,59,142),(50,151,60,141),(101,136,111,126),(102,135,112,125),(103,134,113,124),(104,133,114,123),(105,132,115,122),(106,131,116,121),(107,130,117,140),(108,129,118,139),(109,128,119,138),(110,127,120,137)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 0 | 18 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 0 | 0 | 1 |
| 9 | 11 | 0 | 0 |
| 30 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 24 | 0 | 0 |
| 17 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,18,0,0,9,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,9,1],[9,30,0,0,11,14,0,0,0,0,1,0,0,0,0,1],[1,17,0,0,24,40,0,0,0,0,40,0,0,0,0,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | Q8 | D5 | D10 | D10 | D10 | D10 | D10 | Dic10 | 2- (1+4) | D4×D5 | D4.10D10 |
| kernel | D4×Dic10 | C4×Dic10 | C20⋊2Q8 | Dic5.14D4 | C20⋊Q8 | C20.48D4 | D4×Dic5 | D4×C20 | C22×Dic10 | Dic10 | C5×D4 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_4\times Dic_{10} % in TeX
G:=Group("D4xDic10"); // GroupNames label
G:=SmallGroup(320,1209);
// by ID
G=gap.SmallGroup(320,1209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀